'  C  O  N  CR  ETE  -  STEEL 

CONSTRUCTION 

(DER  EI  S  EN  B  ETO  N  B AU) 
.  '  EAdlL.MORSCH 


FRANKLIN  INSTITUTE  LIBRARY 

PHILADELPHIA,  PA. 


Digitized  by  tlie  Internet  Arcliive 
in  2015 


https://archive.org/details/concretesteelconOOnnors 


Concrete-Steel 
Construction 

(DER  EISENBETONBAU) 


BY 

PROFESSOR    EMIL  MORSCH 

Of  the  Zurich  Polytechnic,  Zurich,  Switzerland 

AUTHORIZED   TRANSLATION   FROM  THE 
THIRD   (1908)  GERMAN  EDITION,   REVISED  AND  ENLARGED 

BY 

E.   P.  GOODRICH 

Consulting  Engineer 


NEW  YORK 

THE  ENGINEERING  NEWS  PUBLISHING  COMPANY 

London:    ARCHIBALD  CONSTABLE  AND  COMPANY,  Ltd. 

1909 


Copyright,  1909, 
BY 

THE  ENGINEERING  NEWS  PUBLISHING  COMPANY 


Entered  at  Stationers'  Hall,  London,  E.C,  1909 


J.  F.  TAPLEY  Co.,  New  York 


PREFACE  TO    SECOND  EDITION 


In  the  absence  of  a  uniform  literature,  and  in  view  of  the  number  of  pro- 
fusely recommended  systems,  the  first  edition  of  this  work,  published  l)y  the 
firm  of  Wayss  &  Freytag  in  1902,  effected  the  purpose  of  familiarizing  those 
interested  in  the  scientific  principles  of  reinforced  concrete  with  all  the  experi- 
mental researches  available  at  that  time.  The  firm  in  question  was  impelled 
to  publish  it  because  systems  based  on  wholly  unscientific  methods  of  calcula- 
tion, and  offering  no  adequate  security,  were  being  pushed  into  recognition  by 
systematic  advertisement,  so  that  the  danger  was  imminent  that  reinforced  con- 
crete would  forfeit  a  large  proportion  of  the  confidence  it  already  enjoyed,  espe- 
cially if  a  few  failures  should  occur. 

More  than  a  year  after  the  publication,  in  conneetion  with  the  first  edition, 
of  information  in  regard  to  reinforced  concrete,  were  published  the  "Leitsatze" 
(Recommendations)  of  the  Verbands  Deutscher  Architekten  und  Ingenieur- 
Vereine,  and  of  the  Deutscher  Beton  Verein,  as  well  as  the  "Regulations" 
(Bestimmungen)  of  the  Prussian  government,  but  they  harmonized  exactly  with 
those  of  the  first  edition.  The  publication  of  the  second  edition  had  another 
purpose.  The  "Leitsatze"  and  the  official  "Regulations"  had  inspired  wide- 
spread confidence  in  the  new  method  of  building,  but  even  the  best  of  direc- 
tions could  not  altogether  obviate  mistakes  and  failures,  where  the  proper 
knowledge  of  the  cooperative  effects  of  the  two  materials — steel  and  concrete 
— was  lacking.  In  addition  to  this,  all  directions  presumed  a  knowledge  of 
approved  rules  of  construction,  as  the  "  Leitsatze  "  could  not  possibly  be  amplified 
into  a  book  of  instructions  on  reinforced  concrete.  This  knowledge  was,  how- 
ever, very  difficult  to  obtain  from  the  class  journals  and  other  literature,  because 
in  these,  all  sorts  of  systems  were  simultaneously  described,  and  conflicting 
opinions  were  also  expressed. 

The  active  part  taken  by  the  firm  of  Wayss  &  Freytag,  as  well  as  the  under- 
signed. Prof.  E.  Alorsch,  in  the  compilation  of  the  preliminary  "Recommenda- 
tions," and  the  interest  they  manifested  in  making  them  final,  caused  them  to 
bring  out  the  present  second  edition,  which  represents  a  complete  revision  of 
the  first  edition,  and  facihtates  the  application  of  the  "  Leitsatze." 

The  general  portion  deals  with  examples  chiefly  relating  to  the  practical 
reinforcement  of  T-beams,  columns,  and  arches,  under  the  most  widely  varied 
loads.  The  succeeding,  and  most  comprehensive  part,  treats  of  the  theory  of 
reinforced  concrete,  covers  exhaustively  the  properties  of  materials,  and  then 

iii 

I8897 


iv 


PREFACE  TO  SECOND  EDITION 


applies  the  theory  in  the  closest  possible  manner  to  the  results  of  the  tests. 
The  author  has  avoided  a  repetition  of  useless  theories  on  reinforced  work,  of 
which  there  is  no  lack.  On  the  other  hand,  he  has  succeeded  in  showing  by- 
means  of  tests  that  the  methods  of  calculation  given  in  the  "Leitsatze'^ 
(which  are  identical  with  those  published  in  the  first  edition)  are  well  founded 
and  useful.  At  the  same  time  the  actual  distribution  of  stress  in  reinforced 
sections  was  thoroughly  studied.  The  firm  of  Wayss  &  Freytag  placed  the 
whole  of  their  experimental  data  (in  great  part  hitherto  unpublished)  at  the  dis- 
posal of  the  author  in  the  preparation  of  the  work.  In  addition  Bach  gave 
the  valuable  results  of  the  tests  conducted  for  the  reinforced  concrete  commission 
of  the  Jubilaumstiftung  der  Deutschen  Industrie,  published  in  the  course  of 
the  current  year,  especially  those  relating  to  adhesion. 

The  third  portion,  covering  the  uses  of  reinforced  concrete,  reviews  the 
most  important  fields  of  its  utilization.  All  the  examples  cited  represent  work 
done  by  the  firm  of  Wayss  &  Freytag,  and,  for  the  most  part,  executed  under 
the  direction  of  the  author  in  his  capacity  as  director  of  the  Technical  Bureau 
of  the  before-mentioned  firm,  selected  from  their  fifteen  years'  experience  in 
reinforced  concrete  work.  This  limitation  of  the  choice  of  examples  is  war- 
ranted, inasmuch  as  all  the  reinforced  construction  work  completed  by  the  firm 
in  question  during  the  past  five  years  has  been  calculated  in  accordance  with 
the  methods  recommended  in  the  "  Leitsatze,"  and  in  accord  with  the  rules  given 
in  the  theoretical  and  general  sections  of  the  book  regarding  construction  work. 

The  field  of  employment  for  reinforced  concrete  is  constantly  widening; 
there  can  therefore  be  no  claim  raised  that  it  has  been  completely  covered;  only 
the  most  important  features  have  been  presented.  But  the  operations  of  this 
single  firm  give  an  excellent  idea  of  the  versatility  of  the  employment  of  rein- 
forced concrete. 

The  firm  is  well  aware  that  the  material  herewith  presented  is  of  service  to 
their  competitors,  but  believe  that  by  a  general  deepening  of  knowledge  of  rein- 
forced concrete,  they  are  rendering  the  most  service  to  the  subject. 

Wayss  &  Freytag. 

Neustadt,  a  d.  Haardt,  November,  1905, 

Professor  E.  Morsch, 

Zurich,  November,  1905. 


PREFACE  TO  THIRD  EDITION 


Owing  to  the  quick  sale  of  the  second  edition,  at  the  request  of  the  publishers 
and  of  the  firm  of  Wayss  &  Freytag,  the  undersigned  undertook  the  preparation 
of  a  third  edition.  Of  the  new  experiments  conducted  by  the  firm  in  the  interim, 
special  attention  must  be  called  to  those  relating  to  shear  in  T-beams  and  those 
made  upon  continuous  Ijeams. 

These  experiments,  in  connection  with  the  recently  published  results  of  the 
tests  undertaken  for  the  Reinforced  Concrete  Commission  of  the  Jubilaumstif- 
tung  der  Deutschen  Industrie,  by  the  Testing  Laboratory  at  Stuttgart,  made  possible 
a  detailed  treatment  of  the  subject  in  question.  Compared  with  the  preceding 
editions  it  is  here  that  the  principal  additions  occur.  In  addition,  the  theo- 
retical chapters  relating  to  flexure  and  bending  with  axial  stress,  were  consider- 
ably extended.  In  the  applications,  the  chapters  on  buildings,  columns,  and 
silos  have  likewise  been  enlarged. 

In  the  preface  to  the  second  edition,  the  grounds  were  given  that  led  to  the 
exclusive  use  of  the  w^ork  of  the  firm  of  Wayss  &  Freytag.  These  reasons  still 
apply  in  regard  to  the  new  edition,  for  most  of  the  examples  referred  to  in  the 
applications  were  made  under  the  author's  direction,  and  he  also  furnished  the 
firm  with  the  suggestions  for  the  new  tests.  The  author  has  also  collaborated, 
as  a  member  of  the  Commission,  in  the  program  of  tests  conducted  by  the  Testing 
Laboratory  at  Stuttgart. 

In  view  of  the  present  general  development  of  reinforced  concrete  the 
standpoint  of  this  work  may  possibly  be  designated  as  one-sided.  It  may  be 
answered  that  the  present  advance  in  the  art  is,  in  large  part,  due  to  the  efforts 
of  the  firm  of  Wayss  &  Freytag,  and  that,  on  the  other  hand,  no  complete 
presentation  of  all  of  the  applications  of  reinforced  concrete  are  contemplated, 
because  the  scope  of  this  work  is  much  too  limited. 

Professor  E.  Morsch. 

Zurich,  November,  1907, 

V 


PUBLISHERS'  NOTE 


Professor  Morsch's  Eisenbetonbau^'  is  probably  the  clearest  exposition 
of  European  methods  of  reinforced  concrete  construction  that  has  yet  been 
published.  It  has  for  some  years  been  a  recognized  standard  in  Europe  and 
has  also  had  a  considerable  demand  in  this  country,  but  the  comparatively 
limited  usefulness  of  the  German  edition  to  American  engineers  prompted  us 
to  make  arrangements  with  Professor  Morsch  for  the  rights  of  translation  and 
publication  of  the  book  in  the  English  language. 

In  the  original  German  edition  there  is  no  division  into  chapters,  but  for 
the  sake  of  clearness  and  system,  and  in  conformity  with  American  custom,  the 
translation  has  been  divided  into  parts,  (i)  The  Theory  of  Reinforced  Concrete, 
and  (2)  The  Applications  of  Reinforced  Concrete,  which  have  been  subdivided 
into  Chapters  and  an  Appendix. 

On  account  of  the  impossibility  of  securing  the  original  drawings  and  photo- 
graphs from  which  to  make  reproductions  for  illustration,  it  was  necessary  to 
import  electros  of  the  cuts  used  in  the  German  book.  Wherever  possible,  the 
wording  of  these  has  been  translated  into  English  and  altered  in  the  cut,  but  in 
many  cases  such  alterations  were  impossible  and  the  German  lettering  has  been 
left. 

The  measurements  used  in  the  German  editions  were  in  the  metric  system 
only;  in  the  translation,  the  metric  system  has  been  retained,  but  the  English 
equivalents  are  given  wherever  measurements  are  quoted,  as  well  as  in  all  tables. 
Furthermore,  a  table  of  metric  and  English  equivalents  has  been  included  at 
the  end  of  the  book. 

It  is  hoped  that  the  efforts  of  the  publishers  to  make  available  to  English- 
speaking  engineers  the  contents  of  this  valuable  work  will  merit  their  approval 
and  appreciation. 

The  Engineering  News  Publishing  Company, 
Book  Department. 

New  York,  November,  1909. 

vi 


CONTENTS 


INTRODUCTION 

PAGE 

CHAPTER  I.  Slabs   4 

T-Beams   8 

Columns   ii 

Arches   14 

PART  I 

THEORY  OF  REINFORCED  CONCRETE 

CHAPTER  n.  Strength  and  Elasticity   15 

Steel   16 

Concrete   18 

Strength  and  Elasticity  of  Concrete   19 

Elasticity  Test  of  Concrete   23 

CHAPTER     111.  Shear,  Adhesion,  Torsion   31 

CHAPTER     IV.  Extensibility   50 

CHAPTER      V.  Compression   59 

CHAPTER     VI.  Simple  Bending   74 

Rectangular  Sections — Slabs   76 

Rectangular  Sections,  Double  Reinforcement   87 

CHAPTER   VII.  Actual  Ultimate  Bending  Tests  of  Reinforced  Slabs  in 

their  Relation  to  Theory   90 

Bending  Tests  of  Concrete  Beams  with  Double  Reinforce- 
ment   93 

CHAPTER  VIII.  Bending  with  Axial  Forces   119 

Bending  with  Axial  Tension   127 

Graphical  Methods  of  Calculation   130 

Method  of  Computation  for  Stage  11a   137 

CHAPTER     IX.  Effects  of  Shearing  Forces   139 

Formulas  for  Shearing  and  Adhesive  Stresses   141 

vii 


viii 


CONTENTS 


PAGE 

CHAPTER      X.  Experiments  Concerning  the  Action  of  Shearing  Forces  ...  151 

CHAPTER     XL  Stuttgart    Experiments    Concerning    Shear,  Continuous 

Members,  etc   174 

Deductions  from  the  Experiments   181 

Shearing  Stresses  in  Beams  of  Variable  Depth   190 

Deformation   192 

Computation  of  Forces  and  Moments   194 

Experiments  with  Continuous  T-Beams   199 

PART  n 

APPLICATIONS  OF  REINFORCED  CONCRETE 

CHAPTER   XIL  Historical   204 

Buildings   209 

Stairs   232 

Arches  in  Buildings   235 

Spread  Footings   241. 

Sunken  Well  Casings   244 

Water-tight  Cellars   247 

Piles   250 

CHAPTER  XIIL  Bridges   256 

With  Horizontal  Members.    Slab  Culverts   256 

Arch  Construction   267 

Reservoirs  -   284 

Silos   287 

APPENDIX 

PRELIMINARY  RECOMMENDATIONS  (LEITSATZE)  FOR  THE 
DESIGN,  CONSTRUCTION,  AND  TESTING  OF  REIN- 
FORCED CONCRETE  STRUCTURES   317 

L  General. 

II.  Building  Preliminaries. 

III.  Checking  Plans. 

IV.  Building  Construction. 

a.  Supervision  of  Work  and  Employees. 
h.  Materials  and  Their  Handling. 

1.  Reinforcement. 

2.  Cement. 

3.  Sand,  Gravel,  and  other  Aggregates. 

4.  Concrete. 

c.  Forms  and  Supports.    Time  of  Removal. 

d.  Protection  of  Structural  Parts. 


CONTENTS 


ix 


V.  Inspection  and  Test  of  Work. 

a.  Tests  during  Erection. 

b.  Tests  after  Completion. 

c.  Duties  of  the  Contractor. 
VI.  Exceptions. 

APPENDIX   TO   THE   FOREGOING   RECOMMENDATIONS  RE- 
GARDING mp:thods  of  calculation  to  be  used 

IN  TESTING  RJ:INF0RCED  CONCRETE  STRUCTURES...  321 

I.  Fundamental  Assumptions. 

a.  ILxternal  Forces. 

1.  Loads. 

2.  Reactions,  Moments,  and  Shears. 

b.  Internal  Forces. 

c.  Safe  Stresses. 

11.  Approximate  Methods  of  Computations. 

a.  Simple  Bending. 

1.  Rectangular  Sections.    Slabs.  ^ 

2.  T-B earns. 

b.  Compression. 

III.  Supports  for  Safe  Loads  . 

IV.  Examples  of  the  Method  of  Computation  for  a  Few  Simple  Cases. 

a.  Simple  Bending. 

1.  Slabs. 

2.  T-Beams. 

3.  Continuous  T-Beams. 

b.  Compression,  Columns. 

REGULATIONS  OF  THE  ROYAL  PRUSSIAN  MINISTRY  OF 
PUBLIC  WORKS,  FOR  THE  CONSTRUCTION  OF  REIN- 
FORCED CONCRETE  BUILDING,  MAY  24,  1907   333 

I.  General. 

a.  Testing. 

b.  Construction. 

c.  Removal  of  Forms. 

11.  Recommendations  for  Statical  Computations. 
a.  Dead  Load. 
*  b.  Determination  of  External  Forces. 

c.  Determination  of  Internal  Forces. 

d.  Permissible  Stresses. 

III.  Methods  of  Calculation,  with  Examples. 

a.  Simple  Bending 

b.  Central  Loading. 

c.  Eccentric  Loading. 

d.  Examples 


CONCRETE-STEEL  CONSTRUCTION 


(Der  Eisenbetonbau) 


CHAPTER  I 

INTRODUCTION 

Reinforced  concrete  (Eisenbeton)  is  the  name  given  to  all  varieties  of 
construction  in  which  are  combined  cement-concrete  and  steel,  in  such  manner 
that  the  two  elements  acting  together,  statically  resist  all  external  forces. 

In  this  connection  it  is  to  be  understood  that  the  concrete  resists  compres- 
sive stresses  principally,  while  the  steel  resists  tensile  ones  in  large  measure — 
that  is,  gives  the  concrete  a  higher  tensile  strength.  In  this  type  of  construction 
many  advantages  and  valuable  properties  result  from  the  combination  of  these 
two  quite  dissimilar  materials.  Buildings  erected  in  this  manner  combine  the 
massiveness  of  concrete  with  the  lightness  of  steel  construction,  and  their  wide 
distribution  and  daily  growth  in  numbers  is  due  to  considerable  economic 
advantages  possessed  by  reinforced  concrete  over  corresponding  work  in  stone, 
wood  or  iron.  Besides  being  cheaper  in  first  cost  than  iron  or  wood,  practically 
all  maintenance  charges  can  be  eliminated  in  reinforced  concrete,  because  of 
the  rational  manner  in  which  use  is  made  of  the  wearing  qualities  of  the  two 
elements.  Another  excellent  property  of  reinforced-concrete  work  is  its  resist- 
ance to  fire.  Because  of  this  quality,  concrete  has  been  employed  for  some 
time  in  building  work,  in  the  shape  of  partitions  and  stairways,  and  for  the  fire- 
proofing  of  steel  beams  and  columns.  Now,  columns  and  beams  are  built  of 
the  same  materials  which  were  formerly  used  simply  for  fireproofing  purposes, 
and  in  this  way  is  secured  a  more  uniform  and  cheaper  fireproof  construction. 

These  several  advantages,  and  the  usefulness  of  reinforced  concrete  for  the 
•structural  parts  of  beams,  columns,  and  floor  slabs,  arise  from  the  following 
fundamental  properties  of  concrete  and  steel  in  combination: 

I.  Steel  Covered  with  Concrete  is  most  Perfectly  Protected  by  it -against 
Corrosion.  This  is  now  a  recognized  fact,  but  it  should  be  added  that  only 
with  relatively  rich  mixtures,  and  with  a  plastic  condition  of  the  concrete  (not 
^arth-moist)  can  there  be  attained  the  intimate  covering  and  adhesion  neces- 
sary to  give  proper  protection.    If  a  leaner  and  drier  mixture  is  employed,  it 


2 


CONCRETE-STEEL  CONSTRUCTION 


is  necessary  to  wash  the  reinforcement  with  cement  grout  just  before  the  deposit 
of  the  concrete,  to  obtain  the  desired  adhesion  and  security  against  rust. 

As  a  proof  of  the  existence  of  this  property  of  protecting  against  rust,  there 
may  be  cited  the  numerous  reinforced-concrete  reservoirs  and  sewers  which 
have  already  stood  for  several  decades  and  as  yet  show  no  signs  of  any  corrosion 
of  the  reinforcement.  Some  examinations  of  twenty-year  old  sewers  showed 
the  steel  absolutely  uninjured  and  of  the  same  color  as  when  it  left  the  rolling 
mill.  Additional  proofs  are  constantly  being  adduced  by  the  repeated  loading 
of  structures,  and  through  the  demolition  of  old  reservoirs  and  floors,  in  none 
of  which  has  ever  been  disclosed  any  corrosion  of  properly  covered  reinforcement, 
even  when  of  considerable  age.  Bauschinger  gives  the  following  report  of  some 
observations  as  to  freedom  from  corrosion  in  several  test  specimens  which  had 
been  broken  in  Ootober,  1887,  and  had  lain  in  the  open  air  till  1892: 

"From  several  slabs,  the  concrete  covering  the  reinforcement  was  knocked 
away  with  a  hammer.  The  chips  broke  only  in  small  pieces  where  the  concrete 
was  struck,  showing  good  adhesion  between  the  steel  and  the  concrete,  and 
the  exposed  reinforcement  was  entirely  free  from  rust,  even  close  to  fractured 
edges. 

"A  tank  was  cracked  and  otherwise  damaged  through  rough  treatment 
during  transportation,  so  that  the  reinforcement  was  partially  exposed.  Natu- 
rally, the  portion  longest  exposed  showed  corrosion,  and  some  rust  was  revealed 
when  the  concrete  was  removed  adjacent  to  an  old  crack.  However,  when  the 
metal  was  exposed  under  an  unbroken,  hard  surface,  no  rust  was  revealed  and 
the  same  adhesion  was  observed  as  in  the  slabs. 

"On  July  23,  1892,  several  fragments  of  floor  slabs  6  to  8  cm.  (2.4  to  3.1  in.) 
thick,  were  examined.  They  had  lain  around  the  end  of  a  sewer,  and  the  pieces 
next  the  entrance  were  most  of  the  time  covered  with  water  which  often  con- 
tained sewage.  According  to  a  statement  of  the  owner,  the  pieces  had  been  in 
place  about  four  years,  and  had  been  purchased  by  him  at  the  sale  of  the  frag- 
ments of  the  tests  made  in  1887.  They  plainly  showed  the  fractured  ends  from 
which  the  reinforcement  stuck  about  5  cm.  (2  in.).  On  one  piece  which  lay 
somewhat  lower  than  the  others,  the  reinforcement  was  scarcely  i  cm.  (0.4  in.) 
beneath  the  upper  surface.  This  upper  layer  was  chiseled  away,  the  concrete 
proving  very  hard  and  adhering  firmly  to  the  steel.  The  latter  was  absolutely 
rustless  to  within  a  distance  of  i  cm.  (0.4  in.)  from  the  fractured  edge."  (See 
Beton  und  Eisen,  No.  IV,  1904,  p.  193.) 

2.  The  Adhesion  between  Embedded  Steel  and  Cement  Concrete  is 
Considerable  and  about  equal  to  the  shearing  strength  of  concrete.  This 
adhesion  can  be  demonstrated  by  direct  experiment,  but  its  presence  is  clearly 
shown  by  the  great  bending  strength  of  reinforced  concrete  slabs  as  compared 
with  those  of  plain  concrete.  This  bending  resistance,  with  reinforcement 
aggregating  1%  of  the  cross-section,  amounts  to  178  kg/cm-  (2532  lbs/in^)  and 
increases  to  247  kg/cm^  (3513  lbs/in^)  with  1.45%  of  reinforcement;  whereas  the 
bending  strength  of  a  plain  concrete  slab  of  similar  section  amounts  at  most 
to  47  kg /cm-  (668  lbs/in^).  If  adhesion  were  lacking,  slabs  with  embedded 
steel  would  show  smaller  bending  strength  than  similar  slabs  without  reinforce- 
ment, because  of  the  diminished  net  concrete  section. 

For  some  time  adhesive  strength  was  assumed  as  40  kg/cm^  (569  lbs/in^) 
as  found  by  Bauschinger,  and  until  lately  its  actual  value  was  considered  unim- 


INTRODUCTION 


3 


portant,  since  adhesion  was  never  taken  into  account  in  making  computations. 
However,  this  point  is  of  great  importance,  and  the  anchorage  of  reinforcing 
rods  should  always  be  investigated. 
Other  tests  will  be  discussed  later. 

With  an  adhesive  strength  of  35  kg/cm-  (498  lbs/in^),  the  length  to  which  a 
rod  must  be  embedded  in  concrete  so  that  its  tensile  strength  (3600  kg /cm-,  or 
51,200  lbs/in^)  is  exceeded  by  the  adhesion  developed,  will  be,  for  a  round 
rod  of 


10  mm. 

diameter, 

26  cm. 

(s  in.) 

(10.2  in.) 

20  mm. 

<  ( 

52  cm. 

(f  in  ) 

(20.4  in.) 

30  mm. 

78  cm. 

in.) 

(30.6  in.) 

and  it  is  seen  that  the  transfer  of  stress  from  the  concrete  to  the  steel,  or  vice 
versa,  may  be  considered  as  proportional  for  shorter  lengths. 

Furthermore,  as  an  additional  precaution  against  slipping  (which  costs  very 
little  extra)  the  ends  of  all  rods  should  be  hooked. 

3.  The  Coefficients  of  Linear  Expansion  by  Heat  of  Steel  and  Concrete 
are  Practically  Identical.  The  coefficients  were  determined  by  Bonniceau 
{Annals  des  ponts  ct  chaiissees,  1863,  p.  181)  for  1°  C.  as 

0.00001235  for  steel  rods,  and 
0.00001370  for  Portland  cement  concrete. 


but  it  is  to  be  understood  that  the  coefficient  for  concrete  is  subject  to  small  varia- 
tions from  differences  in  the  quality  of  the  aggregate. 

Some  experiments  of  Keller  published  in  No.  24  of  the  Tonindustriezeitung, 
1894,  may  be  cited  further.  The  concrete  of  the  test  specimens  consisted  of 
part  gravel,  of  particles  of  20  mm.  (f  in.)  diameter,  and  part  Rhine  sand.  The 
average  coefficients  of  linear  expansion  for  1°  C,  between  —16°  and  +  72°  C, 
were  as  follows: 

Mixture  t:o,  coefficient  0.0000126 
"         1:2,  *'  O.OOOOIOI 

1:4,        ^'  0.0000104 
"       1:8,        "  0.0000095 


The  coefficient  for  steel  is  usually  assumed  as  0.000012. 

Since  the  coefficients  of  expansion  by  heat  are  so  nearly  equal,  the  objection 
formerly  made  against  reinforced  concrete  is  therefore  groundless — that  the 
necessary  adhesion  which  must  exist  between  two  such  dissimilar  materials  as 
compose  it,  would  be  endangered  by  changes  of  temperature.  In  any  case, 
the  temperature  of  thoroughly  encased  steel  cannot  be  far  different  from  that 
of  its  concrete  cover.    Furthermore,  being  poor  conductors,  such  bodies  will 


4 


CONCRETE-STEEL  CONSTRUCTION 


absorb  very  little  heat,  and  this  absorption  will  take  place  only  very  slowly  and 
at  points  directly  exposed  to  temperature  effects.  The  concrete  cover  there- 
fore protects  the  reinforcement  very  effectively  against  temperature  change. 

According  to  official  fire  tests,  a  failure  of  adhesion  which  would  be  danger- 
ous to  strength  does  not  take  place  even  with  large  and  sudden  temperature 
changes  (see  "Das  System  Monier,"  1887,  by  G.  A.  Wayss).  With  usual  dif- 
ferences in  temperature  the  variation  in  expansion  is  compensated  by  small 
internal  stresses  (Zeitschrift  des  O ester r.  Arch-  und  Ingenienr-Vereins,  1897,  No. 

50). 

The  variation  in  volume  of  concrete,  due  to  its  humidity,  has  the  greatest 
influence  upon  the  distribution  of  the  stress  between  the  steel  and  the  concrete. 
Through  experiments,  especially  those  of  the  French  Commission,*  it  has  been 
determined  that  concrete  which  sets  in  air,  shrinks;  while  that  which  sets  under 
water  expands.  General,  accurate  figures  for  the  different  kinds  of  cement, 
and  their  different  mixtures,  cannot  be  given,  although  these  phenomena  are 
worthy  of  more  attention  on  the  part  of  designers  than  they  have  hitherto  received. 

The  several  structural  parts  of  reinforced  concrete  buildings  are  slabs,  T- 
beams,  columns  and  arches — the  characteristics  of  each  of  which  will  first  be 
briefly  described. 


SLABS 

Slabs  are  the  simplest  reinforced-concrete  constructions  built  to  resist  bend- 
ing stresses.  It  is  well  known  that  in  a  slab  simply  supported  at  each  end  and 
centrally  loaded,  the  upper  layers  are  subjected  to  compressive  stresses,  while 
the  lower  layers  are  acted  upon  by  tensile  ones.  Since  the  tensile  strength  of 
concrete  is  much  smaller  than  its  compressive  strength,  the  failure  of  such 
a  concrete  slab  will  take  place  through  exceeding  the  ultimate  tensile  strength. 
It  is  the  province  of  the  added  reinforcement  to  overcome  this  defect,  and  increase 
the  resultant  strength  of  the  structure,  by  carrying  the  major  part  of  the  tensile 
stresses.  The  reinforcement  must  be  designed  so  as  to  have  its  strength  in  a 
proper  ratio  to  the  compressive  strength  of  the  concrete. 

In  slabs  assumed  as  simply  supported  at  the  ends,  the  reinforcing  rods  should 
run  parallel  with  the  lines  of  action  of  the  tensile  stresses,  and  should  lie  as  close 
to  the  bottom  of  the  slab  as  is  consistent  with  proper  protection.  With  good 
mortar,  small  rods  may  be  properly  covered  with  0.5  cm.  (0.2  in.)  of  concrete; 
while  slightly  heavier  material  should  have  at  least  i  cm.  (0.4  in.)  of  covering, 
and  still  larger  rods  should  be  placed  at  greater  distances  above  the  bottom  of 
the  slab.  Usually,  in  addition  to  these  "carrying  rods,"  others  at  right  angles 
to  them,  called  "distributing  rods,"  are  installed.  They  are  primarily  employed 
to  keep  the  carrying  rods  properly  spaced  during  the  construction  of  the  slab, 
and  the  two  series  are  therefore  wired  together  at  points  of  intersection. 

Of  course,  the  number  and  size  of  these  distributing  rods  must  depend  upon 

*  Commission  du  ciment  arme.  Experiences,  rapports,  etc.,  relatives  a  Femploi  du  beton 
arm^.    Paris,  H.  Dunod  et  F.  Pinat,  1907. 


INTRODUCTION 


5 


the  conditions  of  loading  and  support.  They  also  assist  in  distributing  con- 
centrated loads  over  a  larger  carrying  area  of  the  slab. 

If  the  slabs  are  supported  on  four  sides,  the  heavier  carrying  rods  are  laid  in 
the  direction  of  the  shorter  s})an,  and  the  smaller  distributing  rods,  perpendicular 
to  it.  The  section  of  the  carrying  rods  must  vary  as  the  span  and  the  load  to  be 
carried.  Their  spacing  should  be  from  5  to  15  cm.  (2  to  6  in.),  and  it  is  to  be 
noted  that  light  rods,  closely  si)aced,  carry  more  than  larger  rods  with  greater 
spacing.  The  criterion  for  calculating  this  spacing  is  the  unit  adhesive  stress 
on  the  surface  of  the  rods  over  the  supports. 
The  diameter  of  the  distributing  rods  is  usually  5 
to  7  mm.  (t%  to  I  in.)  and  their  spacing  10  to  40 
cm.  (4  to  16  ins.). 

The  distributing  rods  have  another  important 
province  in  cases  where  conditions  are  such  that 
stresses  due  to  temperature  change  are  set  up  in 
the  slab  at  right  angles  to  the  carrying  rods.  In 
such  cases  the  distributing  rods  take  up  the 
stresses  and  thereby  prevent  cracking.  Sometimes 

a  light  system  of  reinforcement  is  installed  near  the  upper  surface  of  a  slab. 
This  is  done  where  absolute  freedom  from  cracking  is  necessary,  and  where 
large  secondary  stresses  are  to  be  expected,  due  to  shrinkage  or  temperature 
change. 

Above,  have  been  considered  only  slabs  freely  supported  at  their  ends.  In 
most  constructions,  however,  a  certain  amount  of  restraint  is  experienced  where 
slabs  are  supported  in  outside  walls,  and  many  slabs  run  continuously  over 
girders  of  rolled  beams  or  of  reinforced  concrete.  Because  of  this  restraint, 
due  to  continuity  of  structure,  the  moment  at  the  middle  of  the  slab  span  is 
reduced,  but  bending  moments  of  opposite  kind  are  produced  over  the  supports, 
and  because  of  this  condition,  reinforcement  must  be  introduced  near  the  tops 
of  the  slabs  in  the  vicinity  of  the  supports,  so  as  to  take  up  the  tensile  stresses 


Fig.  2. 


at  those  points.  In  this  way  is  derived  the  type  of  bent  rod,  originally  used  by 
Monier,  which  corresponds  (in  its  relation  to  the  neutral  axis)  with  the  line  of 
maximum  moments.  A  single  type  of  bent  rods  is  usually  not  sufficient,  since 
moving  loads  must  be  considered.  More  frequently,  both  a  maximum  and  a 
minimum  moment  line  is  involved,  to  which  the  reinforcement  must  correspond. 
Frequently,  too,  it  is  necessary  to  employ  continuous  top  rods,  especially  when 
a  short  span  adjoins  a  long  one.    (See  Fig.  i.) 

Fig.  3  shows  in  detail  the  arrangement  of  reinforcement  employed  in  the 
continuous  slab  of  Fig.  2,  consisting  of  four  spans  supported  between  I-beams. 
The  dotted  line  in  Fig.  4,  which  is  drawn  between  the  two  maximum  moment 
lines,  represents  the  moments  under  conditions  of  perfect  restraint  at  the  ends 


6 


CONCRETE-STEEL  CONSTRUCTION 


and  a  uniformly  distributed  load.    Under  such  conditions  the  moment  is 


24 


in  the  center,  and 


12 


at  the  ends. 


Continuous  reinforced  concrete  floors  between  I-beams  are  usually  con- 
structed with  slightly  arched  ceilings,  the  arches  being  formed  by  constructing 
haunches  down  to  the  lower  flanges  of  the  beams.    The  advantage  of  these 


Fig.  3 


haunches  is  that  for  the  moments  near  the  supports  (which  exceed  those  at  the 
centers)  the  concrete  has  been  so  increased  in  depth  that  no  special  increase  in 
reinforcement  is  necessary.  An  increase  in  the  section  of  concrete  at  the  sup- 
ports is  needed,  if  the  slab  thickness  at  the  center  of  the  span  is  so  thin  as  just 
to  resist  the  compression  at  that  point.  If  this  thickness  were  carried  over  the 
intermediate  supports,  the  concrete  would  be  over-stressed  at  those  points. 
According  to  the  theory  of  continuous  beams  with  variable  section,  because 


^^^^ 


Fig.  4. 


of  the  arch  form  of  the  slab,  a  slight  reduction  results  in  the  moments  at  the 
centers  of  the  spans,  with  a  corresponding  increase  of  those  over  the  supports. 
Since  ample  reinforcement  is  generally  provided  at  the  latter  points,  the  exact 
and  detailed  computation  of  moments  may  be  omitted  in  most  practical  cases. 

In  the  same  manner,  floor  slabs  which  run  continuously  over  reinforced 
concrete  girders  must  be  reinforced.  (See  Fig.  5.)  For  want  of  accurate  knowl- 
edge concerning  the  matter,  no  account  is  taken,  in  either  case,  of  the  torsional 
resistance  exerted  by  the  rolled  steel  or  reinforced  concrete  beams.  Thus,  a 
somewhat  larger  factor  of  safety  is  secured. 


INTRODUCTION 


7 


In  thin  slabs  up  to  about  lo  cm.  (3.9  in.)  thickness,  the  bending  of  the  rods 
should  be  done  with  a  slope  of  1:3.  In  thicker  and  shorter  slabs  the  slope  can 
be  steeper — 1:2  to  1:1^.  It  is  evident,  in  this  connection,  that  in  all  continuous 
slabs,  without  regard  to  an  arrangement  to  fit  the  distribution  of  moments,  so 
much  reinforcement  must  be  bent  that  the  bent  portion  is  able  to  carry  the  whole 
load  of  the  central  portion  of  the  slab  over  into  the  ends,  which  act  as  canti- 
levers, even  though  the  slab  be  cracked  entirely  through  in  the  vicinity  of  the 
bends.    This  rule  is  easy  to  follow,  and  is  the  more  important  the  less  the  amount 


Fig.  5. 


of  straight  reinforcement  and  the  more  the  concrete  is  exposed  to  outside  stresses 
from  shrinkage  and  temperature  change. 

Instead  of  finishing  the  ends  of  the  straight  and  bent  rods  as  hooks,  it  is  evi- 
dent, under  such  circumstances,  that  the  ends  which  lie  next  the  centers  of  the 
slabs  can  remain  straight  and  simply  be  anchored  in  the  zone  of  compression 
of  the  concrete. 

The  number  of  "systems"  of  reinforced  concrete  floors  is  large,  and  new 
''.systems"  are  constantly  being  devised.  In  most  cases,  however,  their  new- 
ness does  not  include  any  improve- 


ments. As  stated  before,  many  systems 
are  at  fault  in  that  no  reinforcement 
is  provided  near  the  upper  surface  over 
the  beams,  as  computations  show 
necessary,  reinforcement  being  used  only  near  the  bottom;  while  others  employ 
a  wrong  distribution  between  the  upper  and  lower  systems  of  rods. 

One  improvement  in  such  floor  systems  aims  at  a  seiDaration,  as  far  as  pos- 
sible, of  the  zones  of  tension  and  compression,  without  essentially  increasing 
the  total  weight  of  the  structure.  This  is  accomplished  by  employing  numerous 
smafl  ribs  separated  by  hollow  blocks  or  grooves  filled  with  light  pumice  con- 
crete.   The  reinforcement  is  placed  in  the  lower  parts  of  the  ribs.    (Fig.  6.) 


Fig.  6. 


8 


CONCRETE-STEEL  CONSTRUCTION 


T=BEAMS 


If  the  hollow  blocks  above  described,  or  the  other  light  fiUing  material,  is 
omitted,  the  floor  construction  consists  of  T-beams  of  concrete  with  the  steel 
enclosed  by  the  stems  of  the  T's.  If  the  ribs  are  arranged  further  apart,  and 
are  built  proportionately  larger,  then  what  was  formerly  the  compression  zone 

must  now  be  treated,  in  ac- 
cordance with  established  rules, 
as  a  restrained  reinforced  con- 
crete  slab    between  beams. 


Fig.  7. 


In  this  way  is  developed  a  construction  in  which  the  slabs  and  beams  combine 
to  form  a  statically  effective  T-section. 

It  is  also  possible  to  design  slabs  and  independent  beams  of  proper  strength 
and  of  simple  rectangular  sections,  but  it  is  clear  that  by  making  the  slabs  carry 
the  compressive  stresses,  a  considerable  economy  is  practised.  The  stressing 
of  the  concrete  slab  in  two  directions  at  right  angles  to  each  other,  is  not  at  all 
hazardous,  and  occurs  in  numerous  other  types  of  construction.  From  a  theo- 
retical standpoint,  a  slab  strengthened  with  ribs  is  more  economical  of  material 
than  a  slab  of  uniform  thickness.  At  a  certain  span,  the  greater  cost  of  instal- 
ling the  ribs  equals  the  saving  in  material,  so  that  T-beams  can  first  be  built 
economically  with  spans  of  between  3  and  4  meters  (10  to  13  ft.). 

Between  the  slabs  and  beams,  naturally  occur  shearing  stresses,  for  the 
transference  of  which  most  builders  arrange  special  vertical  reinforcing  mem- 
bers called  stirrups  (Biigel)  consisting  of  6  to  10  mm.  (J  to  f  in.)  round  rods, 
or  of  thin,  flat  iron.  These  enclose  the  bottom  reinforcing  rods  and  thus  pre- 
vent the  formation  in  the  concrete  of  the  ribs,  of  possible  longitudinal  cracks 


Fig.  9. 


Fig. 


which  might  be  caused  by  the  hooked  ends  of  the  main  reinforcing  rods.  The 
stirrups  thus  increase  the  adhesive  strength,  so  that  it  equals  at  least  that 
employed  in  calculations. 

As  proved  by  tests  which  will  be  described  later,  stirrups  with  none  but 
straight  main  reinforcing  rods  have  only  a  small  effect  on  the  increase  of  the 
shearing  strength  of  the  ribs,  so  that  their  practical  value  consists  in  more  securely 
connecting  the  slabs  and  beams,  and  producing  a  better  distribution  of  the  adhesion. 

So  as  to  secure  the  best  transfer  of  forces  from  one  to  the  other,  the  connec- 
tion l>etween  beams  and  slabs  is  variously  designed,  as  illustrated  in  Figs.  8  to 
10.  By  so  doing,  the  advantage  is  also  gained  of  strengthening  the  slabs  where 
greatest  moments  occur.  With  this  design,  arched  ceilings  between  reinforced 
concrete  beams  are  produced.    (Fig.  8.) 

As  with  flat  slabs,  a  single  low  layer  of  reinforcement  is  not  found  satisfactory, 
especially  if  there  is  any  restraint  at  the  ends,  or,  if  the  beams  are  continuous, 


INTRODUCTION 


9 


over  several  supports.  Similarly,  at  points  of  negative  moment,  steel  must  be 
introduced  near  the  tops  of  the  T-beams,  or  by  carrying  certain  rods  up  and 
over  the  supports. 

Under  certain  load  conditions,  continuous  top  reinforcement  may  be  neces- 
sary, especially  with  unequal  spans.  Furthermore,  at  the  simply  supported  ends 
of  the  slabs  of  heavily-loaded  T-beams,  some  of  the  lower  reinforcement  should 
be  bent  upwards  (at  an  angle  of  about  45°)  so  as  to  take  up  the  shearing  stresses, 
or  rather,  the  diagonal  tensile  stresses  in  the  slabs,  for  which  reinforcement  must 
be  provided.  Since  the  moments  decrease  toward  the  ends  of  the  slabs,  not 
all  of  the  rods  are  necessary  close  to  the  bottom  in  the  vicinity  of  the  supports, 
so  that  a  part  can  advantageously  be  bent  upward. 

The  ribs  are  usually  located  underneath  the  slabs,  but  there  are  also  cases 
in  which  they  may  be  placed  above  them.  One  or  the  other  arrangement  will 
be  employed,  according  to  circumstances. 

Since  the  moments  are  partly  positive  and  partly  negative  for  restrained  and 
continuous  beams,  no  special  advantages  are  gained  with  ribs  located  above 
the  slabs. 


Fig.  II. 


At  the  intermediate  supports,  where  the  greatest  moments  are  found,  the 
compression  occurs  along  the  lower  edge  of  the  beam.  In  order  to  lessen  the 
unit  stress,  the  beam  section  is  increased  at  such  points  by  means  of  a  bracket 
or  knee,  producing  a  slightly  arched  effect.  In  cases  where  the  ribs  are  located 
above  the  slabs  it  is  possible  to  do  without  these  knees,  since  the  whole  width 
of  the  slab  between  ribs  serves  as  a  zone  of  compression. 

The  knees  or  brackets  at  the  intermediate  supports  have  the  added  advan- 
tage of  considerably  reducing  the  unit  shearing  stresses,  partly  because  of  the 
increased  depth  of  beam,  but  principally  because  the  compressive  stresses  along 
the  lower  edges  of  the  beam  at  such  i)oints  act  obliquely  upward  and  thus  equili- 
brate a  part  of  the  diagonal  forces.    (See  Fig.  13.) 

Figs.  II  and  12  respectively,  illustrate  advantageous  arrangements  of  rein- 
forcing rods  for  a  simply  supported  and  a  continuous  T-beam.  Bending  the 
rods  upward  at  the  intermediate  support,  and  anchoring  them  in  the  adjoin- 
ing beam  brings  about  an  economy  in  their  use  in  resisting  the  regular  distri- 
bution of  moments;  and  furthermore,  this  bent  form  increases  the  resistance 
of  the  stem  of  the  T-beam  against  shearing  forces.  Viewed  in  this  light,  it  is 
evidently  to  be  recommended  (with  reference  to  better  anchorage  in  the  concrete 
of  the  ribs),  that  some  at  least  of  the  upper  steel  which  terminates  near  the 


10 


CONCRETE-STEEL  CONSTRUCTION 


intermediate  supports  should  be  bent  obliquely  downward,  as  shown  in  Fig.  14. 
In  that  figure  is  also  shown  how  the  knees  may  be  reinforced  so  as  to  increase 
their  compressive  strength. 

If  the  stem  of  the  T-beam  does  not  afford  enough  room  to  allow  all  the 
reinforcing  rods  to  be  placed  side  by  side,  they  may  be  arranged  in  layers,  in 
which  case  it  is  possible  to  place  the  bent  rods  on  top,  as  shown  in  Fig.  11. 
This  should  be  done  only  in  case  of  necessity,  since  the  rods  are  more  effective 
statically  when  closer  together  than  when  they  are  arranged  in  two  or  more 


Fig.  12. — Reinforcement  and  moment  lines  for  continuous  beams  of  three  spans. 


layers,  because  their  centroid  is  then  lower.  With  continuous  beams  over  spans 
of  varying  lengths,  if  a  long  span  is  fully  loaded  it  may  be  necessary  to  provide 
continuous  top  reinforcement  in  the  adjacent  shorter  spans.  The  amount  of 
restraint  afforded  continuous  beams  by  intermediate  reinforced  concrete  sup- 
ports or  partition  walls,  is  comparatively  small  and  may  well  be  neglected.  The 
same  is  even  truer  in  the  case  of  the  end  supports,  since  positive  restraint  will 
occur  only  in  the  rarest  instances,  where  special  means  have  been  adopted  to 
provide  it. 

At  simply  supported  ends  of  T-beams  care  should  be  taken  to  run  some  of 


INTRODUCTION 


11 


the  lower  rods  straight  over  the  supports.  The  required  number  is  to  be  deter- 
mined by  the  necessary  adhesion. 

.  With  large  spans  the  standard  lengths  of  rods  will  not  suffice,  so  that  welding 
will  be  necessary.     The  weld  should  be 
located  where  the  rod  is  not  fully  loaded, 
which,  in  general,  is  in  a  bend. 

If  a  room  of  given  dimensions  is  to  be 
floored,  it  is  first  divided  into  panels  by  main 
girders,  with  intermediate  supports  if  neces- 
sary. These  girders  are  then  connected  by 
simple  slabs,  or  beams  may  be  introduced 
between  the  girders  so  as  to  diminish  the  slab 
spans.    In  that  case  the  slabs  are  supported 


Fig.  13. 


on  all  four  sides,  and  require  a  correspondingly  light  reinforcement,  especially  in 
a  direction  parallel  with  their  greatest  dimension.  The  principal  reinforce- 
ment is  placed  in  the  opposite  direction,  or  perpendicular  to  the  beams. 

When  both  girders  and  beams  are  employed,  and  the  slabs  are  used  as 
flanges  of  the  girders,  these  slabs  will  be  thrown  into  compression  and  their 


Fig.  14. — Reinforcement  for  an  intermediate  support  of  a  continuous  beam. 


stress  must  be  added  to  their  proper  stresses  from  bending.  For  that  reason 
it  is  recommended  that  only  small  widths  of  slabs  be  used  in  computing  girders, 
and  that  the  slabs  be  constructed  with  haunches,  where  slabs  and  girders  meet. 


COLUMNS 

In  columns,  several  varieties  are  to  be  distinguished.  Some  are  reinforced 
with  vertical  round  rods,  others  with  rolled  shapes  which  are  made  into  rigid 
frames,  and  since  1902  the  spiral  reinforcement  invented  and  patented  by 
Considere  has  been  employed.  Further,  in  the  first  two  varieties,  the  horizontal 
connections  between  the  vertical  pieces  are  of  special  importance  in  connection 
with  the  strength  of  the  column.  Instead  of  temporary  wooden  forms,  rein- 
forced cylinders  or  cement  blocks  can  also  be  used,  the  latter  being  especially 
appHcable  to  bridge  piers.  In  building  work,  the  concrete  columns  take  the 
place  of  cast  or  wrought  iron  ones,  and  must  be  as  small  in  diameter  as  possible. 
Consequently,  the  use  of  a  permanent  shell  is  out  of  the  question. 

By  the  term  "reinforced-concrete  column"  is  usually  understood  one  con- 
taining vertical  round  rods.    Such  a  column  is  constructed  in  the  following  way: 


12 


CONCRETE-STEEL  CONSTRUCTION 


so 


A  concrete  column  of  any  section  contains  a  certain  number  of  vertical  rods 
which  are  placed  close  to  the  surface.    At  certain  points  the  rods  are  fastened 

together  with  horizontal  wire 
tires.  The  whole  reinforce- 
ment thus  forms  a  skeleton, 
which  encloses  the  concrete 
and  prevents  lateral  bulging. 
The  result  is  that  even  in 
long  columns,  ignoring  the 
necessary  safety  against  bend- 
ing, the  strength  of  plain  cubes 
will  be  attained.  The  latter 
is  higher  than  that  of  prisms. 
The  ties  are  placed  from  20 
to  40  cm.  (8  to  16  ins.) 
apart. 

For  a  square  column,  the 
reinforcement  usually  consists 
of  four  rods  located  in  the 
corners,  with  ties  of  7  to  8  mm. 
(approximately  \  to  3%  in.)  wire. 
With  large  dimensions,  eight 
rods  are  used.  (See  Figs.  15 
and  16.) 

The  lower  ends  of  the  ver- 
tical reinforcing  rods  rest  on  a 
grid  of  flat  bars,  so  that  the 
load  carried  by  the  rods  may 
be  distributed  over  a  larger 
area  of  concrete.  This  grid  is 
usually  placed  in  a  separate 
concrete  pedestal,  which  dis- 
tributes the  column  load  over 
a  larger  surface  of  the  foun- 
dation concrete  proper,  corres- 
ponding with  the  lesser  allow- 
able unit  stress  of  the  latter. 
In  columns  which  extend 
through  several  stories  of  a 
building,  the  sections  diminish 
upward,  and  the  rods  have  to 
be  offset  at  each  change  of 
diameter.  Further,  rods  have 
to  be  spliced,  which  can  be 
pipe   over   the   blunt  ends. 


Flat-iron 


Fig.  15. — Base  and  section  of  a  reinforced  concrete 
column. 


short 


piece 


of 


done  simply    by    slipping  a 
(Fig.  17.) 

Greater  resistance  against  bending  is  afforded,  however,  by  lapping  the 


INTRODUCTION 


13 


vertical  rods  from  50  to  80  cm.  (20  to  30  ins.  approximately)  and  by  having 
their  ends  hooked.    (See  Fig.  18.) 

Naturally,  the  column  section  may  be  rectangular,  hexagonal,  octagonal, 
circular,  etc.,  and  the  number  of  reinforcing  rods  can  be  increased  in  propor- 
tion to  the  load.  With  eccentric  loading,  they  should  all  be  placed  on  one  side. 
The  interiors  of  columns  can  also  be  made  hollow  by  enclosing  pipes  in  the  con- 
crete.   These  can  serve  for  rain  leaders,  or  may  contain  gas  or  water  mains. 

The  diameter  changes  to  correspond  with  the  load  to  be  carried,  and  with 
the  factor  of  safety  desired.  It  may  run  from  20  by  20  cm.  (8  hy  8  ins.)  to  70 
by  70  cm.  and  more  (28  by  28  ins.).  The  diameter  of  the  rods  may  vary  from 
14  to  40  mm.  (h  in.  to      ins.  approximately). 


i  i 


Fig.  17.  Fig.  18. 

Splicing  of  rods  in  concrete  columns. 

Columns  of  spirally  reinforced  concrete  designed  by  Considere  have  rela- 
tively light-strength  longitudinal  rods,  while  the  greater  part  of  the  load  is  carried 
by  a  spiral  wrapping  which  encloses  the  longitudinal  rods  and  the  concrete  core 
within  them.  This  spiral  affords  great  resistance  against  the  bulging  of  the 
concrete  under  load.  The  spirals  should  be  covered  by  concrete,  so  that  the 
best  shape  for  such  a  column  is  round,  octagonal,  or  hexagonal.  The  first  pub- 
lication by  Considere  concerning  his  "beton  frette,"  or  hooped  concrete,  was 
in  "Genie  Civil,"  in  November,  1902.  His  investigations  on  concrete  cylinders 
with  spiral  reinforcement  disclosed  an  efficiency  2.4  times  greater  for  the  rein- 
forcing material  than  when  used  simply  as  straight  rods,  and  the  strength  of 
the  concrete  was  increased  to  800  kg/cm^  (11,400  lbs/in^),  or  about  quadrupled. 
Practical  applications  are  already  quite  numerous  and  are  especially  useful 
in  cases  where  it  is  necessary  that  a  very  heavily  loaded  column  should  have 
a  small  diameter. 


14 


CONCRETE-STEEL  CONSTRUCTION 


ARCHES 

The  reinforcement  for  small  arches  can  be  determined  in  the  same  manner 
as  for  simple  slabs.  Since  no  bending  moments  act  on  an  arch  with  a  parab- 
olic profile  and  uniform  loading,  a  system  of  lightly  interwoven  reinforcement 
near  the  soffit  is  usually  sufficient.  Usually,  however,  such  simple  reinforcement 
is  not  enough,  a  second  layer  near  the  upper  surface  extending  from  the  abut- 
ments over  the  haunches  being  needed.  In  bridge  arches  which  are  subjected 
to  variations  of  load,  reinforcement  is  introduced  throughout  near  both  the 
upper  and  lower  arch  surfaces. 

Reinforced  concrete  arches  have  the  advantage  over  arches  of  plain  concrete 
that  the  reinforced  arch  can  withstand  tensile  stresses  as  well  as  compressive 
ones.  For  short  spans  it  is  thus  possible  to  secure  reinforced  arches,  which  make 
full  use  of  the  compressive  strength  of  the  concrete.  Under  such  conditions, 
arches  of  much  less  thickness  are  secured  than  when  non-reinforced  concrete 
is  used,  the  thickness  of  which  for  short  spans  must  be  made  so  great  as  to  pre- 
vent the  appearance  of  appreciable  tensile  stresses. 

In  arches  of  larger  span,  properly  designed  to  meet  the  conditions  involved, 
tensile  stresses  do  not  occur,  and  the  question  of  reinforcement  lessens  in  impor- 
tance since  it  does  not  change  the  unit  compressive  stresses  enough  to  compensate 
for  its  employment.  With  wide  spans,  the  profile  of  the  arch  is  of  considerable 
importance,  so  that  the  unit  compressive  strength  of  the  concrete  shall  not  be 
exceeded;  while,  with  short  spans,  and  with  the  introduction  of  reinforcement, 
the  form  of  the  arch  can  be  freely  chosen  within  certain  limits.  Cases  often 
occur  in  building  work  where  the  form  of  an  arch  must  be  selected  for  archi- 
tectural reasons  not  corresponding  at  all  with  the  statical  conditions,  and  only 
a  reinforced  arch  can  be  employed. 

Just  as  in  slabs,  so  in  arches — lateral  reinforcement  is  employed,  which  serves 
the  same  purpose  as  the  distributing  rods  in  slabs,  and  is  similarly  designated. 
The  upper  and  lower  systems  of  reinforcement  in  arches  are  held  in  the  desired 
relative  positions  by  means  of  wire  ties. 

Besides  round  rods,  rolled  shapes  are  sometimes  used  in  arches  (Melan  sys- 
tem). Then  the  arch  is  composed  of  a  series  of  parallel  ribs  which  are  entirely 
embedded  in  concrete.  In  floor  arches  and  other  small  structures,  the  ribs  are 
T-bars,  rails  or  wide-flanged  I-beams,  and  are  connected  with  each  other  only 
at  the  points  of  support.  With  larger  spans  and  deeper  ribs,  the  latter  are  built 
as  lattice  girders,  and  bars  are  run  between  them.  These  serve  mainly  as  sup- 
ports for  the  arch  forms. 


PART  I 


CHAPTER  II 
THEORY  OF  REINFORCED  CONCRETE 
STRENGTH  AND  ELASTICITY 

In  the  early  stages  of  the  development  of  reinforced  concrete,  its  l^uilders 
had  at  hand  no  recognized  methods  of  calculation,  and  Monier  and  Franc^ois 
Coignet  erected  their  work  solely  by  practical  instinct  and  experience.  Of  late, 
a  real  rivalry  has  developed  in  the  production  of  new  theories  concerning  reinforced 
concrete,  and  thejr  authors  have  been  anxious  to  explain  the  particular  excel- 
lence inherent  in  a  combination  of  steel  and  concrete,  with  reference  to  their 
combined  statical  action.  Practice  has  here  been  far  ahead  of  theory.  The 
principal  question  in  controversy  has  been  whether  the  tensile  strength  of  the 
concrete  in  bending  should  be  considered.  Among  practical  builders  this  ques- 
tion was  really  decided  at  the  start,  and  decided  against  its  inclusion,  because 
absolutely  no  attention  is  paid  to  it  and  the  steel  is  stressed  to  the  maximum 
safe  limit.  The  tensile  strength  of  the  concrete  is  entirely  ignored.  On  this 
assumption  was  based  the  first  method  of  theoretical  computation  of  slabs, 
devised  by  Koenen  (Government  architect)  in  Berlin  in  1886,  and  his  method 
has  been  used  by  the  majority  ever  since. 

Theoretical  investigators,  unfamiliar  with  the  practical  side  of  concrete  con- 
struction, usually  considered  the  tensile  strength  of  the  concrete,  and  some  even 
went  so  far  in  the  older  methods  as  to  assume  the  elasticity  in  tension  and  com- 
pression as  equal.  Later,  the  modulus  of  elasticity  in  tension  was  accepted  as 
smaller  than  that  for  compression,  and  a  parabola  was  assumed  as  the  stress- 
strain  curve.  Finally,  the  stress  curve  for  concrete  in  tension  was  found  by 
Considere's  investigations  to  be  a  straight  line  parallel  with  that  of  the  steel. 
It  is  evident  that  with  such  assumptions,  results  are  obtainable  which  appear 
extremely  accurate  to  the  several  authors;  but  long  formulas  are  not  attractive 
to  practical  builders,  and  in  this  connection  it  is  to  be  observed  that  the  employ- 
ment of  a  parabola  for  the  stress-strain  curve  is  actually  less  accurate  than  the 
use  of  a  straight  line,  because  a  certain  amount  of  violence  must  ])e  used  if  the 
stress-strain  curve  is  forced  into  parabolic  form.  But,  ignoring  this  point,  such 
methods  of  calculation  do  not  provide  the  desired  degree  af  safety,  and  may 
even  become  actually  dangerous  if  too  small  a  percentage  of  reinforcement  is 
used. 

It  is  not  the  object  of  this  book  to  give  a  review  of  all  proposed  methods  of 
calculation.  This  would  be  useless,  and  furthermore,  the  methods  of  checking 
designs  contained  in  the  "Vorlaufige  Leitsatze  fiir  Eisenbetonbauten "  (Tentative 

15 


16 


CONCRETE-STEEL  CONSTRUCTION 


Recommendations  concerning  Reinforced  Concrete  Construction)  published  in  1904 
by  the  Verband  Deutscher  Architekten-  und  Ingenieurverein  and  the  Deutscher 
Beton-Verein,  and  in  the  "  Bestimmungen  fiir  die  Ausfiihrung  von  Konstruktionen 
aus  P^isenbeton  bei  Hochbauten"  (Regulations  for  the  Execution  of  Constructions 
in  Reinforced  Concrete  in  General  Building  Work),  issued  by  the  Prussian  govern- 
ment, are  identical  with  those  contained  in  the  first  edition  of  this  book  (1902). 
The  new  requirements  of  the  French  Ministry  of  Public  Works  of  October  20, 
1906,  for  posts  and  telegraphs,  also  contain  the  same  assumptions  and  methods 
of  computation.  Therefore,  here  will  be  discussed  only  the  theory  above 
described,  which  has  been  proved  best  by  several  years  of  trial  and  in  a  large 
number  of  constructions.  Since  the  first  edition,  the  results  of  numerous 
experiments  have  been  secured  which  test  the  accuracy  of  these  methods  of 
calculation  and  especially  explain  the  importance  of  shear  in  T-beams. 

Methods  of  calculation  will  therefore  be  found  in  close  connection  with  the 
results  of  experiments.  In  no  other  subject  is  it  more  important  to  rely  as 
completely  on  the  results  of  tests,  if  disagreeable  experiences  are  to  be  avoided, 
since  the  present  knowledge  concerning  reinforced  concrete  is  at  best  imper- 
fect and  liable  to  surprises.  Before  turning  to  the  methods  of  calculation, 
which  are  very  simple,  a  review  will  be  made  of  the  strength  and  elastic  prop- 
erties of  steel  and  plain  concrete,  so  that  the  formulas  may  be  more  susceptible 
of  daily  use. 


The  properties  of  steel  (wrought  iron  or  steel)  are  well  known  to-day.  In 
calculations  relative  to  steel  construction,  the  relation  between  stresses  and 
strains  is  assumed,  and  the  limiting  ratio  will  never  be  exceeded  in  actual  load- 
ing.   Furthermore;  the  tensile  strength  is  the  same  as  the  compressive  strength, 


and  the  elastic  behavior  is  the  same  under  tensile  and  compressive  stresses.  As 
to  the  modulus  of  elasticity  and  the  safe  working  stress,  opinions  do  not  differ 
materially.  Usually,  wrought  iron  in  the  form  of  rods  is  employed  *  for  rein- 
forcement. In  Table  I  some  results  are  given  for  ordinary  material  from  stock. 
In  it,  d  represents  the  diameter  of  the  machined  test  specimen,  not  of  the  rod 
from  which  the  specimen  was  prepared. 

In  special  locations,  such  as  arch  bridges,  the  steel  reinforcement  can  be 


STEEL  (EISEN) 


Fig.  19. 


*  In  Europe. — Trans. 


THEORY  OF  REINFORCED  CONCRETE 


17 


used  in  the  form  of  rolled  shapes  or  of  lattice  girders.  The  American  expanded 
metal  (Fig.  19)  invented  by  Golding,  made  by  stamping  and  bending  sheet  metal, 
has  been  highly  recommended  for  the  reinforcement  of  slabs.  Any  required 
strength  can  be  obtained  by  change  of  thickness,  and  size  of  mesh.  However, 
when  iLsing  expanded  metal  one  does  not  have  as  easy  a  means  of  adapting  the 
design  to  the  variation  of  the  moments,  as  with  the  use  of  round  rods,  so  that 
expanded  metal  can  be  used  only  for  simple  slal)s.  The  lighter  grades  are  used 
for  ornamental  plaster  beams  of  various  kinds.  In  stamping  the  meshes  from 
the  sheet,  the  material  experiences  a  heavy  stress,  and,  since  the  strength  and 
abihty  of  ingot  iron  to  stretch  are  damaged  by  stamping,  the  sheets  must  be 
annealed  in  order  to  remove  this  defect. 


Table  I 

RESULTS  OF  TESTS  ON  ROUND  IRON  MADE  BY  THE  TESTING  LABORA- 
TORY OF  THE   ROYAL  TECHNICAL  HIGH  SCHOOL,  STUTTGART 


Stretch  in 

Reduc- 

Diameter 

Elastic  Limit 

Tensile  Strength 

Modulus  of  Elasticity 

Length 

tion  in 

of  10  d 

Area 

mm. 

in. 

kg/cm2 

lbs/i.i2 

kg/cm2 

lbs/in  2 

kg/cm2 

lbs/in2 

% 

% 

10 

0-39 

2994 

42590 

4178 

59430 

2192000 

31 180000 

10 

0-39 

3026 

43040 

4182 

59480 

2143000 

30480000 

26.4 

66.9 

10 

0-39 

3104 

44150 

4123 

58640 

2140000 

30440000 

27.0 

69. 1 

10 

0-39 

3117 

44330 

4234 

60220 

2172000 

30890000 

24.8 

66.9 

10 

0-39 

3038 

43210 

4329 

61570 

71 .0 

15 

0-59 

2710 

38550 

3810 

54190 

21 I 6000 

30100000 

27.  2 

55-3 

15 

0-59 

2725 

38760 

4146 

58970 

2150000 

30580000 

30-0 

71.7 

15 

0-59 

2627 

37370 

3870 

55050 

2140000 

30440000 

26.4 

55-6 

15 

0-59 

2938 

41790 

4124 

58660 

2133000 

30340000 

28.0 

71.6 

15 

0-59 

3277 

46610 

4610 

65570 

30-0 

53-7 

20 

0.79 

2650 

37690 

3940 

56020 

2184000 

31060000 

30-3 

64-4 

20 

0.79 

2166 

30810 

3790 

53910 

2165000 

30790000 

31.2 

64.0 

20 

0.79 

2681 

38130 

3991 

56760 

2161000 

30740000 

30.4 

64-4 

20 

0.79 

2627 

37360 

3845. 

54690 

2177000 

30960000 

31.2 

63-6 

In  America  various  forms  of  reinforcement  are  employed,  all  of  which  are 
designed  to  prevent  slipping  of  the  rod  in  the  concrete.  In  the  Ransome  rod 
(Fig.  20),  this  is  secured  by  twisting  the  square  steel  bar;  in  the  Johnson  bar, 


Fig.  20. 


elevations  on  the  surfaces  of  the  rods  are  produced  in  the  rolling;  and  the 
Thacher  or  knotted  bar  is  provided  with  swellings,  while  maintaining  a  con- 
stant sectional  area.    These  "knots"  may  well  have  the  desired  effect  when 


18 


CONCRETE-STEEL  CONSTRUCTION 


the  rod  is  anchored  in  a  large  mass  of  concrete,  but  they  will  act  in  an  opposite 
manner  in  the  small  stems  of  T-beams,  especially  at  their  bottom.s,  where  they 
will  have  a  splitting  effect  and  thus  cause  premature  failure  of  bond.  It  will 
be  shown  later  that  the  adhesion  in  the  case  of  ordinary  round  rods  with  hooked 
ends  is  ample  to  transfer  all  actual  stresses,  and  furthermore,  the  arrangement 
of  the  principal  reinforcement  may  be  so  designed  with  respect  to  the  shearing 
stresses  that  no  occasion  should  arise  to  make  up  any  deficiency  through  the 
use  of  those  costly  special  bars. 

CONCRETE 

For  reinforced  concrete  work  only  rich  mixtures  of  fine-grained  materials 
should  be  employed.  Practically,  only  with  rich,  wet  concrete,  will  the  neces- 
sary adhesion  and  rust  prevention  be  secured,  because  only  then  will  the  tamp- 
ing force  enough  grout  against  the  reinforcement  to  completely  coat  it.  This 
coating  of  grout  adheres  to  the  concrete  in  spite  of  cracks  and  even  rupture  between 
the  concrete  and  the  steel,  and  forms  the  real  rust  preventative,  as  can  be  demon- 
strated. When  using  drier  and  poorer  concrete,  it  is  important  to  coat  the  rein- 
forcement with  cement  grout  immediately  before  depositing  the  concrete. 

The  sand  aggregate  exerts  a  great  effect  in  determining  the  quality  of  the 
concrete.  With  the  cement  it  forms  the  mortar,  ond  on  the  strength  of  this 
mortar  depends  the  strength  of  the  concrete.  The  strength  of  the  latter  is  usually 
somewhat  greater  than  when  no  gravel  is  used.  In  the  "  Mitteilungen  iil^er 
Druckelastizitat  und  Druckfestigkeit  von  Betonkorpern  mit  Verschiedenem  Was- 
serzusatz"  (Communication  Concerning  the  Compressive  Strength  and  Elasticity 
of  Concrete  Specimens  with  Different  Admixtures  of  Water),  Stuttgart,  iqo6, 
pages  II  and  14,  the  following  figures  are  given,  which  are  of  interest  in  this 
connection: 

The  compressive  strength  of  m.ortar  taken  from  a  i :  2J:  5  mixture,  amounted,  in 

28  days  100  days 

to  294  kg/cm2  (4182  lbs/in2)  ^32  kg/cm^  (4722  lbs/in^), 

while  the  strength  of  the  corresponding  earth-moist  concrete  of  1:2^:5  mixture  was 

225  kg/cm^  (3200  lbs/iii2)  321  kg/cm^  (4566  lbs/in2). 

Similarly,  for  a  1:4:8  mixture  (pages  10  and  13),  the  results  were 

Mortar  280  kg/cm^  (3982  lbs/in^)  258  Kg/cm-  (2670  lbs/in^. 
Concrete  230  kg/cm^  (3271  lbs/in^)     254  kg/cm^  (2513  lbs/in2). 

The  "  Leitsatze  "  of  the  Verbandes  Deutscher  Architekten-  und  Ingenieurverein 
recommended  that  in  the  composition  of  concrete  for  reinforced  work,  the  mor- 
tar contain  sand  of  graded  sizes  of  particles  up  to  7  mm.  (0.28  in.),  and  be  mixed 
not  poorer  than  1:3.    Further,  that  the  addition  of  gravel  or  stone  chips  in 


STRENGTH  AND  ELASTICITY  OF  CONCRETE 


19 


quantities  up  to  that  of  the  sand  was  permissible.  The  size  of  the  gravel  or 
stone  should  be  l)et\veen  7  mm.  (0.28  in.)  and  25  mm.  (o.q8  in.). 

Of  cement,  only  the  best  Portland  should  l)e  used  corresponding  at  least 
with  the  "Normen"*  since  not  enough  experience  has  been  obtained  concerning 
other  cements,  es])ecially  as  to  their  action  on  reinforcement. 

Under  certain  conditions,  ])umice  may  advantageously  l)e  used  as  the  j)rin- 
cipal  aggregate  of  concrete.  On  account  of  its  smaller  strength,  j)umice  con- 
crete can  only  be  used  for  light  slab  construction,  princi{)ally  roofs,  where, 
besides  the  advantage  of  its  lighter  weight,  it  also  has  that  of  insulating  against 
temperature  changes.  Although  pumice-concrete  is  ])rinci])ally  used  in  arches, 
between  steel  beams,  it  can  also  be  used  in  the  slabs  of  reinforced  floors,  pro- 
vided the  beams  are  made  of  gravel  concrete.  Pumice-concrete  is  usually  made 
with  river  sand  as  i)art  of  the  aggregate. 

STRENGTH  AND  ELASTICITY  OF  CONCRETE 

Compressive  Strength. — The  resistance  which  concrete  offers  to  crushing 
is  quite  varial)le,  and  changes  with  the  proportions  of  the  mixture  and  with  the 
properties  of  the  sand,  gravel,  and  broken  stone,  as  well  as  with  the  tamping 
during  making.  The  form  and  size  of  the  test  specimen  also  influences  the 
apparent  strength.  The  compressive  strength  per  square  centimeter  decreases 
when  the  section  of  the  specimen  is  enlarged.  The  apparent  strength  is  espe- 
cially dependant  upon  the  ratio  of  the  height  of  the  specimen  to  its  base. 
When  this  ratio  is  small  (as  in  mortar  joints)  the  strength  is  considerable. 
But  when  the  height  is  several  times  the  diameter  of  the  1)ase,  failure  will  occur 
along  a  diagonal  plane,  because  the  shearing  strength  has  been  exceeded,  and 
the  compressive  strength,  which  is  not  involved,  appears  small  when  the  break- 
ing load  is  divided  by  the  area  of  section.  The  compressive  strength  of  concrete 
cubes  is  called  the  "cubic  strength"  (Wurfelfestigkeit)  of  concrete,  and  is  usually 
assumed  as  the  allowable  compressive  strength  in  reinforced  work,  because  in 
such  constructions,  diagonal  shearing  is  prevented  by  the  use  of  proper  rein- 
forcement. 

As  to  the  increase  of  strength  with  age,  some  very  interesting  tests  are  avail- 
able. They  were  made  in  connection  with  the  erection  of  the  bridge  over  the 
Danube  at  Munderkingen.  With  i  part  cement,  2J  parts  sand,  and  5  parts 
pebbles,  mixed  wet,  the  test  cubes  20  cm.  (7.8  in.)  on  each  edge  developed  the 
stresses  shown  in  Table  IL 

Table  II 

LONG  TIME  COMPRESSIVE  STRENGTH  TESTS  OF  CONCRETE 

After  7  days  an  average  compressive  strength  of  202  kg/cm"  (2873  lbs/in^). 

After  28  days  an  average  compressive  strength  of  254  kg/cm"  (3613  lbs/in^). 

After  5  months  an  average  compressive  strength  of  332  kg  'cm^  (4722  Ibs/in^j. 

After  2  years,  8  months  an  average  compressive  strength  of  520  kg/cm^  (7396  lbs/in^). 

After  9  years  an  average  compressive  strength  of  570  kg/cm^  (8107  lbs/in^). 

Lately,  discussion  has  turned  much  to  the  question  of  earth-moist  or  plastic 
concrete.    As  plastic  concrete  is  here  understood  it  contains  50  per  cent  more 

*  German  standard. — Trans. 


20 


CONCRETE-STEEL  CONSTRUCTION 


water  than  necessary,  so  that  it  can  be  placed  in  thicker  layers  and  be  brought 
to  a  proper  consistency  by  a  less  number  of  blows  of  the  tamper  than  can  moist 
concrete.  To  solve  the  problem  as  to  whether  moist  or  plastic  concrete  was. 
the  better,  a  large  number  of  experiments  were  made  at  the  Testing  Labora- 
tory of  the  Technical  High  School  of  Stuttgart  on  the  compressive  strength 
and  elasticity  of  different  proportions.  The  results  of  the  tests,  published  by 
Bach  *  are  of  considerable  value.  Even  by  these  the  question  is  not  conclu- 
sively answered,  since  with  exactly  the  same  materials  the  above  described 
specimens,  which  were  made  in  Ehingen  and  in  Biebrich,  gave  variously 
divergent  results.  While  the  specimens  from  Ehingen  almost  invariably  gave 
substantially  higher  results  for  the  plastic  concrete,  the  specimens  prepared  in 
Biebrich  showed  a  superiority  for  the  moist  concrete,  but  within  two  years  the 
plastic  concrete  increased  as  much  in  strength  as  did  the  moist.  The  use  of 
the  moist  concrete  requires  particularly  expert  workmanship  and  rigorous 
inspection,  but  even  then  involves  the  troubles  incident  to  defective  work.  On 
the  other  hand,  a  considerable  security  is  obtained  with  regard  to  the  uniformity 
of  the  mass  when  plastic  concrete,  that  is,  such  as  has  an  excess  of  water,  is  used. 
In  reinforced  concrete  work,  plastic  concrete  is  especially  valuable,  since  tamp- 
ing is  often  almost  impossible  through  several  layers  of  reinforcement. 

Prismatic  specimens,  like  Fig.  21,  on  which  elasticity  tests  were  made,  gave 
the  following  compressive  strengths,  (each  result  is  the  average  of  three  obser- 
vations; mixture,  i  cement  to  3  gravel  and  sand;  plastic): 

After  3  months,  172  kg/cm^  (2446  lbs/in^) 
After  2  years,     308  kg/cm^  (4381  lbs/in^) 

The  strength  of  reinforced  concrete  buildings,  therefore,  increases  with  time,, 
so  that  one-fifth  of  the  cubic  strength  at  an  age  of  twenty-eight  days  may  well 
be  assumed  as  the  safe  working  stress.  According  to  the  "  Leitsatze,"  under 
ordinary  weather  conditions,  at  an  age  of  twenty-eight  days  the  concrete  should 
develop  a  compressive  strength  in  30  cm.  (12  ins.  approximately)  cubes,  of  180- 
200  kg/cm^  (2560-2845  lbs/in^).  If  this  strength  is  not  developed  with  any 
particular  sand  when  mixed  in  mortar  proportions  of  i  to  3,  then  more  cement 
is  to  be  added.  Moreover,  the  i :  3  mortar  mixture  is  to  be  considered  the  extreme 
limit,  especially  with  regard  to  the  securing  of  ample  protection  against  rust. 

Tensile  Strength.  The  results  of  tensile  tests  are  more  variable  than 
those  of  compression.  All  the  conditions  which  affect  the  apparent  compres- 
sive strength,  affect  the  tensile  strength  as  well,  and  the  shape  and  size  of  the 
test  specimen  is  of  even  more  importance. 

In  the  majority  of  cases,  tensile  tests  are  made  on  mortar  specimens,  that  is, 
on  bodies  composed  only  of  cement  and  sand,  and  are  prepared  only  to  afford 
a  test  of  the  cement.  Few  tests  on  regular  concrete  specimens  have  ever  been 
made.    The  latter  give  lower  results  than  do  specimens  made  of  mortar,  as  is 

*  "  Mitteilungen  iibcr  die  Herstellung  von  Petonkorpern  mit  Verschiedenem  Wasserzusatz, 
sowie  iiber  die  Druckfestigkeit  und  Druckelastizitat  derselben,"  Stuttgart,  1903.  Konrad  Witt- 
wer.  (Report  Concerning  the  Manufacture  of  Concrete  Specimens  with  varying  Percentages, 
of  Water,  together  with  their  Compressive  Strength  and  Elasticity.)  The  second  edition  (igo6) 
contains  the  experiments  on  specimens  two  years  old. 


STRENGTH  AND  ELASTICITY  OF  CONCRETE 


21 


shown  by  the  experiments  made  in  connection  with  some  elasticity  tests  at  the  Testing 
Laboratory  in  Stuttgart,  the  specimens  for  which  are  illustrated  in  Fig.  21. 

The  results  contained  in  Table  III  are  averages  of  three  tests  of  specimens 
made  of  Heidelberg  cement  and  Rhine  sand  and  gravel,  mixed  wet: 


Table  III 
TENSILE  STRENGTH  OF  CONCRETE 


Mixture 

1:3 
1:3 
1:4 


Age 
3  months 

2  years 

3  months 


Tensile  Strength 
12.6  kg/cm^  (179  lbs/in-) 
15.5  kg/cm^  (220  lbs/in^) 
9.2  kg/cm^  (130  lbs/in^) 


"X 


l..._.Jt 


Even  on  similar  specimens  the  results  are  quite  variable,  as  is  shown  by  the 
fact  that  the  number  15.5  is  the  average  of  8.8,  15.8,  and  22.0. 

Elasticity  of  Concrete. — Just  as  it  is  impossible  to  assign  a  definite  value 
to  the  breaking  strength,  so  it  is  impossible  to  do  so  for  the  modulus  of  elasticity 
of  concrete,  since  all  the  above  mentioned  points  influence  the 
elasticity  as  well  as  the  strength.  For  this  reason  the  results 
obtained  by  different  observers  cannot  be  compared,  and 
therefore  it  is  necessary  to  make  special  tests  in  practical 
cases  or  to  select  results  made  under  comparable  conditions. 

Experiments  concerning  the  elastic  deformation  of  Portland 
cement  concrete  under  pressure  have  been  made  by  Durand- 
Claye,*  by  Bauschinger,  and  by  the  committee  on  arches 
of  the  Oesterr.  Ingenieur-  und  Architektenverein,  etc.;  but 
the  most  accurate  and  best  known  are  those  made  by  Bach. 

All  former  tests  w^re  defective  in  that  they  employed 
specimens  of  too  small  dimensions.     Further,  no  distinction  Fig.  21. 

was  made  between  elastic  and  permanent  deformations. 
This  point  was  first  brought  out  by  Bach  in  his  experiments  for  the 
Wiirtt.  Ministerialabteilung  fiir  Strassen  und  Wasserbau,  in  1895."}*  His 
cylindrical  specimens  were  25  cm.  (y.8  in.)  in  diameter,  and  i  meter  (39.4 
in.)  long.  The  shortening  in  a  length  of  75  cm.  (29.5  ins.)  was  measured  at 
two  diametrically  opposite  points.    The  experiments  were  conducted  as  follows: 

A  load  corresponding  to  8  kg/cm-  (113.8  lbs/in^)  was  brought  to  bear  on 
the  specimen  and  then  removed.  This  operation  was  repeated  several  times, 
until  only  pure  elastic  deformation  resulted.  The  load  was  then  increased  to 
16  kg/cm^  (226.6  lbs/in^),  and  the  same  process  of  loading  and  unloading 
repeated  until  the  maximum  permanent  set  for  this  load  had  been  attained.  In 
this  manner  the  operation  was  continued,  and  with  each  increment  the  total 
deformation,  elastic  deformation,  and  permanent  set  were  measured.  Curves 
were  drawn  to  represent  the  values  thus  obtained.  A  definite  elastic  limit  was 
not  disclosed  by  these  curves;  rather,  from  the  start  the  shortening  seemed  to 
increase  with  the  stress. 

In  specimens  made  with  Blaubeurer  cement,  a  straight  line  can  be  substi- 
tuted for  the  stress-strain  curve  up  to  stresses  of  about  40  kg/cm^  (569  lbs/in^). 

*  Annales  des  ponts  et  chaussees,  1888. 

f  Zeitschrift  des  Vereins  Deutscher  Ingenieure,  1895-1897. 


22 


CONCRETE-STEEL  CONSTRUCTION 


The  deformation  curves  found  by  Bach  are  so  regular  that  they  may  be 
represented  by  an  exponential  equation,  the  relation  between  the  compression 
and  the  stress  being  such  that 

E=a 

where  E  is  the  deformation  in  unit  length,  o  the  corresponding  stress,  and  a 
and  m  are  coefficients  which  depend  upon  the  properties  of  the  material.  Sim- 
ilar relations  have  been  deduced  for  sandstone,  granite,  cast  iron,  etc.,  for  all 
materials  in  which  no  constant  proportionality  exists  between  the  stresses  and 
the  strains,  and  in  which  the  tensile  and  compressive  elasticities  differ  considerably. 

The  equations  of  Table  IV  *  have  been  deduced  for  several  different  mixtures, 
but  they  are  not  correct  for  all  brands  of  cement: 

Table  IV 

EXPONENTIAL  EQUATIONS  OF  STRESS-STRAIN  CURVE  OF  CONCRETE 

I  cement:  2i  sand:!;  gravel,  ii= —         o•^■^^  ("^^r  '  coefficient  for  inches  and  lbs.) 

^  ^       '       298000  ^5676100 

I  cement:  25  sand:^  stone,  £=  o'^'^^,  (  coefficient  for  inches  and  lbs.) 

457000  ^9190500  ^ 

I  ^  I  X 

I  cement:  3  sand,  E=  ■  o^'^^,  (-  coefficient  for  inches  and  lbs.  I 

315000  ^0520300  ^ 

I  cement:  li  sand,  E=- — -- —  o^'^^,  ( —  coefficient  for  inches  and  lbs.) 

356000  ^7567200 

Considerable  information  concerning  the  compressive  elasticity  of  much 
tamped  concrete  of  different  mixtures  and  degrees  of  humidity  is  to  be  found 
in  the  above  mentioned  "  Mittelungen  liber  die  Herstellung  von  Betonkorper," 
etc.,  of  Bach,  1903  and  1906. 

The  dearth  of  elastic  tests  on  such  concrete  as  is  used  in  reinforced  con- 
struction work,  and  the  comparatively  few  tests  which  had  been  made  on  the 
elasticity  of  concrete  in  tension  for  the  arch  committee  of  the  Osterr.  Ingen-.  und 
Arch.-Verein,  by  Grut  and  Nielsen,  led  to  the  making  of  some  further  tests  on 
the  elasticity  of  concrete  in  compression  and  tension,  at  the  Testing  Laboratory 
of  the  Royal  Technical  High  School  of  Stuttgart. 

Specimens  like  those  illustrated  in  Fig.  21,  were  made  of  Mannheimer  Port- 
land cement  and  Rhine  sand  and  gravel.  The  aggregate  consisted  of  about 
3  parts  sand  of  o  to  5  mm.  (o  to  0.2  in.)  grains,  and  2  parts  of  gravel  of  5  to  20 
mm.  (0.2  to  0.78  in.)  pebbles. 

The  results  are  shown  graphically  in  Figs.  22  to  25  inclusive,  and  are  also  given 
in  Tables  V  to  VII.  The  numbers  are  always  the  averages  of  three  tests.  Six 
specimens  were  prepared  of  each  of  the  mixtures,  1:3,  1:4,  and  1:7,  with  8  per 
cent  and  14  per  cent  of  water,  one-half  being  tested  in  compression  and  the  other 
half  in  tension.  The  measured  length  was  350  mm.  (13.8  ins.).  The  repetition 
of  load  was  omitted,  so  that  the  experiments  would  not  take  so  long  and  be  so 
tedious,  and  to  produce  an  equivalent  result  at  each  step,  the  load  was  maintained 
for  three  minutes.  The  age  of  the  specimens  was  quite  uniform,  viz.,  80  to  90 
days.  Consideration  is  here  given  only  to  the  i :  3  and  i :  4  mixtures,  because  the 
results  obtained  from  the  i :  7  mixture  were  of  less  value  compared  with  the 
other  two,  and  because  such  proportions  are  not  used  for  reinforced  concrete. 

*  In  metric  units. — Trans. 


STRENGTH  AND  ELASTICITY  OF  CONCRETE 

ELASTICITY  TESTS  OF  CONCRETE 

Table  V 
1:3  MIXTURE 


T 

K 

Unit 

Stress 

8%  Water 

14%  Water 

Vletric 

.g/cm2 

lbs/in  2 

Deformation 
in 

Millionths 

E 

Deformation 
in 

Mililonths 

E 

Metric 

English 

Metric 

English 

[61-3 

871.9 

255 

240000 

3413000 

293 

209000 

2973000 

49.0 

697.0 

198 

247000 

3513000 

227 

216000 

3072000 

36.8 

523-4 

143 

257000 

3655000 

165 

222000 

3158000 

30.6 

435-2 

117 

261000 

3712000 

135 

227000 

3226000 

c 
_o 
'5! 

24-5 

348-5 

92 

266000 

3783000 

104 

235000 

3342000 

18.3 

260.3 

67 

273000 

3883000 

76 

241000 

3428000 

a 
S 

15-3 

217.6 

55 

278000 

3954000 

62 

246000 

3499000 

0 

12.2 

173-5 

43 

284000 

4039000 

48 

254000 

3613000 

9-  2 

130.8 

32 

287000 

4082000 

36 

260000 

3698000 

6.1 

86.8 

21 

290000 

41 25000 

23 

265000 

3769000 

.  3-0 
0 

42.7 
0 

10 

300000 

4267000 

II 

272000 

3869000 

r  1.6 

22.8 

6 

267000 

3798000 

7 

230000 

3271000 

c 

3-1 

44.0 

13 

238000 

3385000 

15 

207000 

2954000 

.9 

i  • 

4-6 

65-4 

20 

230000 

3271000 

23 

200000 

2845000 

6.2 

88.1 

28 

221000 

3143000 

32 

194000 

2759000 

7-7 

109.4 

38 

203000 

2887000 

44 

175000 

2489000 

I  9.2 

130.8 

47 

196000 

2788000 

T 

ensile  Streng 

th 

T 

ensile  Streng 

th 

12.6  (179.2 

) 

10.5  (149-3) 

Table  VI 
1:4  MIXTURE. 


Unit  Stress 

8%  Water 

14%  Water 

Metric 

English 

Deformation 

E 

Deformation 
in 

Millionths 

E 

in 

Millionths 

Metric 

English 

Metric 

English 

61.3 

871.9 

290 

21 1000 

3001000 

360 

170000 

2418000 

49-0 

697.0 

225 

218000 

3101000 

276 

177000 

2518000 

36-7 

522.0 

163 

225000 

3  200000 

198 

185000 

263 1 000 

30.6 

435-2 

133 

230000 

3271000 

160 

I 91 000 

2716000 

c 
_o 

24-5 

348.5 

104 

235000 

3342000 

124 

198000 

2816000 

cn 
QJ 

l-i  ■ 

18.3 

260.3 

76 

241000 

3428000 

90 

203000 

2887000 

a 
S 

15-3 

217.6 

62 

247000 

3513000 

73 

210000 

2987000 

0 
0 

12.2 

173-5 

49 

250000 

3556000 

58 

215000 

3058000 

9-2 

130.8 

36 

257000 

3655000 

42 

219000 

3 II 5000 

6.1 

86.8 

23 

265000 

3769000 

27 

226000 

3214000 

-  3-0 
0 

42.7 

0 

II 

273000 

3883000 

12 

250COO 

3556000 

r  .6 

22.8 

6 

266000 

3783000 

6 

250000 

3556000 

0 

3-1 

44-1 

13 

240000 

3414000 

14 

221000 

3143000 

4.6 

65-4 

21 

224000 

3186000 

22 

200000 

2845000 

6.2 

88.2 

31 

200000 

2845000 

32 

I 94000 

2759000 

I  7.8 

no. 9 

41 

190000 

2702000 

T 

ensile  Streng 

th 

I 

ensile  Streng 

th 

9-2  (130-8) 

8.8  (125  2) 

24 


CONCRETE-STEEL  CONSTRUCTION 


When  deformations  are  taken  as  ordinates  and  stresses  as  abscissas,  the  curves 
of  Figs.  22  to  25  are  obtained: 


14%  water  14%  water 

Figs.  22-25. — Stress-strain  curves  for  concrete. 


STRENGTH  AND  ELASTICITY  OF  CONCRETE 


25 


The  deformation  curves  are  quite  regular  in  shape.  The  tensile  strength  of 
large  concrete  specimens  is  always  considerably  less  than  of  octagonal  mortar 
ones,  since  the  latter  can  be  compacted  much  better  than  can  larger  ones.  Con- 
cerning the  percentage  of  water  used,  it  is  to  be  noted  that  the  specimens  were 
molded  in  water-tight  cast-iron  forms;  that  the  sand  and  gravel  was  not  abso- 
lutely dry,  and  that  the  addition  of  14  per  cent  of  water  (especially  with  the 
poorer  mixtures)  proved  superabundant — a  condition  not  reached  in  practice 
even  with  plastic  concrete.  Measurements  of  deformations  cannot  be  carried 
as  near  to  the  ultimate  strength  as  is  to  be  desired,  because  of  the  danger  of 
damaging  the  measuring  instruments. 

Just  as  the  ultimate  strength  of  concrete  increases  with  age,  so  does  the  modulus 
of  elasticity.  This  can  be  seen  from  the  experimental  results  of  Table  VII  obtained 
on  specimens  two  years  old  mixed  1:3,  with  14  per  cent  of  water.  The  results 
of  tests  on  three-month  old  specimens  are  also  given  for  comparison. 


Table  VII 
ELASTICITY  TESTS  OF  OLD  CONCRETE 


Unit  Stress 


Kg/cm2 

lbs/in2 

'  86.0 

1223. I 

73-7 

1048.2 

61.3 

871.9 

1 

49.0 

697.0 

XT. 

36.8 

523-4 

a 

30.6 

435-2 

6 
0 

24-5 

348.5 

u 

18.3 

260.3 

12.2 

173-5 

.  6.1 

0 

0 

■  1.6 

22.8 

3-1 

44-1 

4-6 

65-4 

c 

6.2 

88.2 

0 

"tr,  < 
C 

7-7 

109.5 

9.2 

130.8 

10.8 

153-6 

12.3 

174.9 

I  13-8 

196.3 

Three  Months  Old 


Deforma- 
tion in 
Millionths 


293 
227 

165 
135 
104 

76 
48 
23 

7 

15 
23 
32 
44 


Metric 


209000 
216000 
222000 
227000 
235000 
241000 
254000 
265000 

230000 
207000 
200000 
194000 
175000 


English 


2973000 
3072000 
3158000 
3229000 
3342000 
3428000 
3613000 
3769000 

3271000 
2944000 
2845000 
2759000 
2489000 


Tensile  strength. 
10.5  (149-3) 


Two  Years  Old 


Deforma- 
tion in 
Millionths 

E 

Metric 

English 

334 

257000 

3655000 

280 

263000 

3741000 

229 

268000 

381 2000 

180 

272000 

3869000 

132 

278000 

3954000 

109 

280000 

3983000 

87 

283000 

4025000 

64 

286000 

4068000 

42 

290000 

41 25000 

20 

305000 

4330000 

4-7 

340000 

4836000 

9-8 

316000 

4495000 

14.8 

31 1000 

4423000 

20.0 

310000 

4409000 

25.0 

308000 

4381000 

30-3 

303000 

4310000 

35-5 

303000 

4310000 

40.8 

301000 

4281000 

46.2 

298000 

4239000 

Remarks 


Tensile  strength. 
15.8  (224.7) 


The  stress-strain  curve  for  the  two-year  old  concrete  is  shown  in  Fig.  26. 

Bending  Strength  of  Concrete. — The  tensile  strength  of  rectangular  con- 
crete beams  calculated  from  actual  bending  tests  carried  to  rupture,  by  means 
of  Navier's  formula,  is  always  about  twice  the  value  obtained  from  direct  tension 
tests.    Several  results  of  bending  tests  on  plain  concrete  beams  will  first  be  given. 


26  CONCRETE-STEEL  CONSTRUCTION 

and  then  the  theoretical  explanation  of  this  seeming  contradiction  will  be  dis- 
cussed. 


Fig.  26. — Stress-strain  curves  for  concrete  3  months  and  2  years  old. 


Experiments  by  Hanisch  and  Spitzer. 

Not  only  were  the  bending  strengths  of  the  slabs  determined,  but  the  tensile 
and  compressive  strengths  were  also  ascertained,  of  specimens  carefully  cut  from 
the  broken  slabs.  The  mixture  was  1:34,  the  clear  span  1.50  meters  (59  in.), 
the  width  of  slab  60  cm.  (23.6  in.),  and  the  age  268  days.    (See  Table  VIII.) 

The  explanation  of  the  seeming  contradiction  has  to  be  sought  in  the  phe- 
nomenon of  the  variation  of  the  modulus  of  elasticity  and  its  difference  for  ten- 
sion and  compression.  Consequendy,  Navier's  formula  cannot  be  used  except 
for  comparative  purposes,  the  computed  extreme  tensile  stress  given  by  it  being 
too  high. 

Prof.  W.  Ritter,  of  Zurich,  has  given,  in  Part  I  of  his  "  Anwendung  der 
'Graphischen  Statik,"  1888,  a  graphical   method  of  computing  stresses  which 
exceed  the  elastic  limit,  applicable  to  all  materials  of  which  the  deformation  dia- 
gram is  curved,  as  is  that  of  cast  iron,  the  relative  behavior  of  which  is  very 
similar  to  that  of  concrete. 


STRENGTH  AND  ELASTICITY  OF  CONCRETE 


27 


Table  VIII 


COMPARISON  OF  COMPRESSIVE,  TENSILE,  AND  BENDING  STRENGTHS 


No 

Thickness 

Concentrated 
Live  Load 

Dead  Load 

Compressive 
Strength 

Tensile 
Strength 

Bending 
Strength, 

cm. 

in 

kg. 

lbs. 

kg. 

bs. 

Metric. 

Engl'h. 

Metric. 

Engl'h. 

Metric. 

Engl'h. 

I 

2 

3 
4 
5 
6 

7-8 
8.0 

lO.O 
lO.O 

3-  0 

4-  5 
4-5 
3-1 
3-9 
3-9 

8oo 
1400 
1500 

700 
1 200 
1 200 

1764 

3086 
3307 
1543 
2644 
2644 

170 
240 
240 

175 
210 
210 

375 
529 
529 
385 
463 
463 

296 

329 
256 

314 
352 
300 

4210 
4680 

3641 
4666 
5007 
4267 

29 

24 
27 

23 
20 

29 

412 
341 
384 
327 
284 
412 

54-6 
43-2 
46. 1 

49-1 
46. 2 

49-1 

777 
614 
656 
698 

657 
698 

Average.  . . . 

308 

4381 

25 

356 

48.0 

6S3 

The  stresses  might  also  be  computed  with  the  help  of  the  exponential  equa- 
tion of  the  stress-strain  curve,  as  was  explained  by  Carling  in  the  Zeitschrift 
des  Oesterreich.  Ingenieur-  und  Architektenverein,  1898.  Assuming  the  elastic 
properties  assigned  to  granite  by  Bach,  Carling  computes,  with  the  help  of 
the  exponential  law,  the  location  of  the  neutral  axis  in  a  rectangular  section, 
the  corresponding  maximum  tensile  and  compressive  stresses,  and  the  relation 
between  depth  of  beam  and  moment  for  assumed  tensile  stresses.  But  since 
the  exponential  law  applies  only  to  low  stresses,  it  cannot  be  em.ployed  in  com- 
putations of  conditions  near  rupture. 

In  the  same  volume  of  the  above  mentioned  Zeitschrift,  Spitzer  gave 
a  method  of  calculation  for  beams  of  materials  possessing  a  variable  deforma- 
tion coefficient,  which,  while  only  approximate,  is  applicable  to  all  beam  shapes, 
and  for  which  a  knowledge  only  of  the  stress-strain  curves  for  tension  and  com- 
pression is  required. 

The  simplest  explanation  of  the  high-bending  strengths  of  concrete  is  found 
in  the  graphical  method  first  above  mentioned,  which  is  as  follows: 

If  Navier's  hypothesis  is  assumed,  as  is  here  done,  in  accordance  with  vv-hich 
plane  sections  before  bending  are  supposed 
to  remain  so  after  flexure,  then  the  deforma- 
tions are  represented  in  Fig.  27  by  the  Hne 
DD'  and  the  different  stresses  by  the  line 
EOE'.  Since  the  ordinates  are  proportional 
to  the  deformation,  the  curve  EOE'  is  none 
other  than  the  experimentally  determined 
stress-strain  curve. 

Fig.  28  shows  this  line,  which  can 
also  be  taken  to  represent  the  stress- 
distribution  in  a  beam  of  rectangular  cross-section.  The  area  within  the  curve 
above  the  neutral  plane  shows  the  total  compressive  stress,  and  the  area  below 
it  is  the  tensile  stress.    Since  no  external  horizontal  forces  act  on  the  beam,  in 


28 


CONCRETE-STEEL  CONSTRUCTION 


every  section  the  total  compression  must  equal  the  total  tension.  That  means 
that  the  areas  OAB  and  OCD,  above  and  below  the  neutral  axis,  must  be  equal. 
Abscissas  above  and  below,  which  intersect  the  deformation  curve  so  as  to  pro- 
duce equal  areas,  therefore  indicate  corresponding  maximum  tensile  and  com- 
pressive stresses.  Each  compressive  stress  corresponds  with  a  perfectly  definite 
tensile  stress.  If  5j  and  5^  are  the  centroids  of  the  areas  OAB  and  OCD,  then 
the  moment  of  the  internal  stresses  is  equal  to  Dy=Zy,  in  which  y  is  the  dis- 


D  Tensile 
Strength 


Fig.  28. 


tance  between  the  centroids.  This  moment  must  be  equal  to  that  of  the  exter- 
nal forces.  When  a  certain  edge  stress  above  or  below  is  assumed,  the  moment 
can  be  expressed  as  a  function  of  (as  is  also  the  case  with  the  exponential 
law).  If  the  ultimate  tensile  strength  is  assumed  as  the  lower  edge 
stress,  the  maximum  possible  moment  for  non-reinforced  conditions  will  be 
obtained. 

If  the  deformation  curve  is  extended  beyond  the  ultimate  tensile  stress,  as 
is  done  in  Fig.  28,  then  are  obtained  the  corresponding  edge  stresses  given 
in  Table  IX  for  the  specimens  described  on  page  22  of  a  1:3  mixture. 


STRENGTH  AND  ELASTICITY  OF  CONCRETE 


29 


Table  IX 

CORRESPONDING  EDGE  STRESSES  IN  CONCRETE  BEAMS 


Compression 

3.5  Kg/cm2 

5-3 

7-2 

9-4 
20.8 
26. 2 


Tension 
3.1  Kg/cm2 
4.6 
6.2 
7.7 

12.6 


Further,  there  is  obtained  at  the  point  of  rupture,  with  0^  =  12.6  and  for  unit 
width, 

D=Z  =  SAh; 
y  =  0.6411 
M  =  5.4Xo.64X/^^=3.45^^  * 

From  this  moment,  the  edge  stresses  are  found  by  Navier's  formula  to  be 

tT=— ==-^2"~=3.45X6  =  2o.7  kg/cm2, 

whereas  the  actual  stresses  are  12.6  on  the  tension  side  and  26.2  on  the  compres- 
sion side. 

Three  actual  bending  tests  of  the  above  mentioned  mixture  gave  an  average 
of  21.4  kg/cm^  for  the  bending  stress  computed  by  Navier's  formula.  This  is 
in  accord  with  the  value  expected  from  the  deformation  curve  computation.  In 
other  words,  it  may  be  used  as  a  partial  check,  since  the  bending  tensile  stress 
shown  by  it  is  entirely  different  from  that  secured  in  true  tension  tests. 

Specimens  when  about  three  months  old  were  tested  to  rupture  with  a  center 
load,  and  the  bending  strengths  given  in  Table  X  were  developed  according  to 
Navier's  formula: 

Table  X* 

COMPARISON  OF  BENDING  AND  TENSILE  STRENGTHS 


Mixture  

1:3 

1:4 

1:7 

Per  cent  of  water.  . . . 

8 

14 

8 

14 

8 

14 

Bending  strength.  . . 

21.4 

23-2 

16. 1 

16.7 

13-3 

12.8 

Tensile  strength  

12.6 

10.5 

9.2 

8.8 

4.4 

5.5 

The  specimens  had  a  length  of  i  meter  (39.37  in.),  a  width  of  15  cm.  (6  ins.), 
and  a  height  of  20  cm.  (7.87  in.).  They  were  mixed  with  Mannheimer  Port 
land  cement  and  Rhine  sand  and  gravel. 

*  Metric. — Trans. 


30 


CONCRETE-STEEL  CONSTRUCTION 


The  bending  strength  of  concrete  is  often  used  in  connection  with  the  com- 
pressive strength  as  a  test  of  the  quality  of  the  material,  since  tests  of  it  are  easier 
to  make  than  tensile  ones,  which  latter  depend  largely  on  the  degree  of  accuracy 
with  which  the  load  is  applied  at  the  exact  center  of  the  specimen.  So  long  as 
the  fact  is  kept  in  mind  that  Navier's  formula  gives  results  good  only  for  com- 
parative purposes,  and  that  the  actual  tensile  stresses  are  only  about  half  those 
shown  by  it,  that  method  can  conveniently  be  used. 


CHAPTER  III 


THEORY  OF  REINFORCED  CONCRETE 

SHEAR,  ADHESION,  ETC. 

Shearing  and  Punching  Strength  of  Concrete  (Schub-  und  Scherfestig- 
keit). — The  great  importance  played  by  shearing  forces  in  reinforced  concrete 
construction,  and  a  study  of  the  results  of  other  tests,  led  to  the  making  of  the 
following  series  of  experiments,  partly  by  the  writer  and  partly  by  the  Testing 
Laboratory  of  the  Royal  Technical  High  School  at  Stuttgart.  The  experiments 
disclosed  a  marked  difference  between  the  qualities  of  shear  and  punching 
resistance      Schubfestigkeit  "  and  "  Scherfestigkeit  "). 

As  is  known,  there  exists  in  every  section  of  a  homogeneous  beam  loaded 
like  those  shown  in  Figs.  29  and  30,  normal  stresses  o  and  shearing  stresses  t, 


Fig.  29.  P'lG.  J. . 


which  combine  to  form  two  inclined  mutually  perpendicular  principal  stresses, 
so-called,  viz.: 

and 

2     \  4 

the  directions  of  which  are  found  from 

2X 

tan  20;  =  

a 

If  it  is  understood  that  between  any  adjacent  sections  no  external  concen- 
trated forces  act  on  the  beam,  the  shearing  stresses,  which  are  to  be  computed 

OS 

according  to  the  formula  t=—^,  occur  in  pairs,  and  at  every  point  the  horizon- 
tal shear  r  is  equal  to  the  vertical  shear.  If  nothing  but  shearing  stresses  act 
in  any  two  adjacent  sections,  so  that  (7  =  0  (as  happens  in  a  cylinder  subject  only 
to  torsion),  then  any  rectangle  ABCD     (Fig.  31)  will  be  deformed  by  the  pairs 

*  Formed  by  differential  portions  of  two  adjacent  sections,  AD  and  BC. — Trans. 

31 


32 


CONCRETE-STEEL  CONSTRUCTION 


of  stresses  into  a  rhomboid,  in  which  the  diagonal  AC  has  been  lengthened, 
and  BD  shortened.  The  principal  stresses  are  then  o^^+z  and  o^^  =  —  r, 
and  the  angle  a  =45°.  These  values  appear  directly  from  the  rectangular  form 
of  the  figure  ABCD.  If  there  is  also  to  be  considered  the  influence  of  the  lateral 
dilation,  it  is  evident  that  for  this  dilation,  due  to  the  corresponding  stresses  on 
the  material  in  the  proper  directions, 


\  m/ 


or  with   w  =  4, 


the  allowable  stress  t=o.8o  a^,  a  value  frequently  employed  in  steel  construction 
and  one  found  by  experiment. 

t 

D 

.1  4 

D 


A 


Fig.  31, 


In  distinction  from  the  types  of  loading  of  Figs.  29  and  30  is  that  of  Fig.  32. 
In  the  former,  only  shearing  stresses  (Schubspannungen)  were  supposed  to  act, 
that  type  being  distinguished  from  other  cases  by  the  condition  that  the  beam 
is  subject  only  to  flexure  and  consequently  deflects.  The  other  variety  is  the  case 
of  pure  shear.*  This  differs  from  the  foregoing,  both  spoken  of  as  shear,  in  that 
no  bending  takes  place  and  the  external  force  is  here  theoretically  applied  only 
on  a  single  section;  while  before,  it  was  constant  through  several  adjoining 
sections  (or  with  a  uniform  load,  varied  only  slightly  from  one  to  another).  It 
is  thus  evident  that  pure  shear  is  scarcely  possible  in  practical  work. 


Fig.  32.  Fig.  33.  F1G.34. 

The  action  of  concrete  amply  justifies  a  distinction  between  beam  shearing 
stresses  and  pure  shearing  or  punching  stresses  (Schub-  und  Scherspannungen) , 
since  they  give  entirely  different  surfaces  of  rupture  and  offer  different  resistances. 

To  obtain  a  relation  between  the  compressive,  tensile,  and  punching  strengths, 
one  may  imagine  the  resistance  to  shear  to  be  similar  to  that  offered  by  a  series 

*  Best  illustrated  by  ibe  action  of  a  punch. — Trans. 


SHEAR 


33 


of  small  teeth,  Fig.  33,*  along  the  infinitesimal  faces  of  which  compressive  and 
tensile  forces  act  in  oblique  but  mutually  perpendicular  directions.  The  hori- 
zontal components  of  these  forces  must  balance  among  themselves,  and  the 
vertical  components  must  equal  the  total  shearing  force  S.  Or,  in  other  words, 
the  shear  ct  in  the  vertical  section  of  a  tooth  (Fig.  34)  is  the  resultant  of  the 
two  normal  forces  ba^  and  aa^,  and  must  pass  through  their  point  of  inter- 
section, which  determines  the  perpendicularity  of  the  faces  of  the  teeth.  Because 
of  the  condition  that  a  rupture  of  this  series  of  teeth  can  occur  only  when  the 
compressive  stresses  and  the  tensile  stresses  a,  simultaneously  reach  their 
ultimate  values,  a  definite  shape  is  imposed  upon  the  right  triangle  abc  and  a 
definite  relation  must  exist  between  the  compressive,  tensile,  and  shearing  strengths. 
In  the  triangle  of  forces 

c2  t^  =  a^af+b^(7^^. 
The  equation  of  the  horizontal  components  gives 

bo^^b^ao^-, 

or, 

b^G^  =  a^o^, 

which,  in  connection  with  the  first  equation,  gives, 

c2  f=b^     o^  +  a^     a^=a^  a^{a^+b^) 

from  which 

The  theoretical  maximum  pure  shearing  strength  would  therefore  be  the 
geometrical  mean  of  the  tensile  and  compressive  strengths. 

In  an  absolutely  homogeneous  material  with  equal  tensile  and  compressive 
strengths,  t  would  equal  a,  or  with  regard  to  lateral  dilation  there  is  obtained 


In  the  case  of  actual  tests  of  wrought  iron  and  steel,  the  strength  in  pure 
shear  equals  0.7  to  0.8  of  the  tensile  strength,  thus  developing  equally  large 
shearing  and  torsional  strengths  (compare  Bach,  "  Elastizitat  und  Festigkeit  "). 
With  concrete,  however,  of  which  the  tensile  strength  is  not  as  large  as  the  com- 
pressive strength,  tests  show  that  the  shearing  strength  is  considerably  larger 
than  the  tensile  one,  and  close  to  the  theoretical  value  t=-Va^  a^^. 

Experiments  |  concerning  Pure  Shear  in  Concrete  with  the  Arrangement 
shown  in  Fig.  36. — The  18  by  18  cm.  (7  by  7  in.)  prismatic  concrete 
specimens  were  fixed  on  one  side  in  a  Marten  testing  machine,  with  cast-iron 
plates  above  and  below,  so  that  the  space  between  the  two  upper  plates  corresponded 

*  Figures  32  to  43  are  loaned  by  the  "Schweizer  Bauzeitung,"  where  they  were  first  pub- 
iished  by  the  author. 

t  These  and  the  following  described  experiments  were  made  by  the  author. 


34 


CONCRETE-STEEL  CONSTRUCTION 


accurately  with  the  width  of  the  lower  plate.  When  the  load  was  applied  on 
the  non-reinforced  specimens,  a  crack  a  first  show^ed  itself  in  the  middle,  run^ 
ning  from  top  to  bottom.  This  was  doubtless  caused  by  a  bending  of  the  speci- 
men. However,  the  load  on  the  machine  could  yet  be  considerably  increased, 
and  only  then  did  the  load  take  full  bearing  on  the  edges  of  the  plates,  as  is  neces- 
sary in  order  to  obtain  the  real  shearing  strength. 

I.  Test  on  three  concrete  specimens,  mixed  1:3,  with  14  per  cent  of  water 
18  by  18  cm.  (7  by  7  in.)  in  section,  age  2  years,  Fig.  35. 


Fig.  35. — Shear  test. 


The  bending  crack  a  appeared  at  a  load  P  =  s  tonnes  (11,000  lbs.),  but  the 
load  was  increased  to  P  =  40  t.  (88,000  lbs.)  when  shearing  along  crack  b  took 
place.  In  the  second  specimen,  the  bending  crack  appeared  atP  =  iot.  (22,000 
lbs.)  and  the  shearing  took  place  at  P  =  38t.  (83,600  lbs.),  while  the  third  speci- 
men sheared  at  P  =  5o  t.  (110,000  lbs.).  On  the  assumption  of  an  equal  dis- 
tribution of  P  between  the  two  sections  to  be  sheared,  the  shearing  strengths 
of  the  three  specimens  result  as  shown  in  Table  XI. 

Table  XI 
SHEARING  STRENGTH 

^=^^^-^'-^  ^g/^™'  (^^79  lbs/in^) 
lo  X  lo 

^=^^«=58.7  kg/cm^  (835  lbs/in^) 

^=  '0^^^=77-2  kg/cm2  (1C98  lbs/in^) 
ioXlo 

Average    65.9  kg/cm^  (937  lbs/in^) 

Tests  of  three  specimens  of  each  kind,  and  of  the  same  age  and  mixture, 
74  cm.  (29.12  in.)  high  and  18  by  18  cm.  (7  by  7  in.)  in  section,  like  Fig.  21,. 


SHEAR 


35 


broken  at  the  Testing  Laboratory  of  the  Technical  W\gh  School  at  Stuttgart, 
gave  the  following  average  values: 


Tensile  strens:th 


8.8  +  15.8  +  22.0  1,0         „     .  . 

Oz=^  ^  =  15-5  kg/ cm-  (220  lbs  ni-), 


Compressive  strength     =  =  t,oS  kg/cm^  (5405  lbs/in^). 


In  accordance  with  the  theory  described  above,  the  limit  of  shearing  strength 
would  be 


(Td=V  15.5X308  =  69  kg/cm-  (981  lbs/in-), 

while  the  observed  strength  was  65.9  kg/cm^  (937  lbs/in-). 

2.  Test  with  18  by  18  cm.  (7  by  7  in.)  con- 
crete prisms,  months  old,  and  1:4  mixture 
with  14  per  cent  of  water.  The  aggregate  con- 
sisted of  3  parts  sand  of  o  to  5  mm.  (o  to  0.2 
in.)  grains,  and  2  parts  of  gravel  of  5  to  20  mim. 
(0.2  to  0.78  in.)  pebbles,  and  was  also  of  the  same 
quality  as  the  other  specimens.  The  arrangement 
is  illustrated  in  Fig.  36. 

Specimen  i:    Bending   crack   in   the  middle 
at  P  =  i5  t.  (33,000  lbs.);  sheared  at   P  =  25  t. 
^5  (SSjOoo  Hjs.).    If  a  uniform  distribution  of  stress 

(Dimensions  in  cm.)  is  assumed,  the  unit  shearing  strength  will  be 


f-  

.  «  : 

 M  

i  ^ 

*<.-  — 

 —  >• 

'^TiyTs^^^-^  (549  lbs/in^). 


Specimen  2  gave  /  =  4i.7  kg/cm-  (593  lbs/in^), 
Specimen  3         /  =  31.0  kg/cm^  (441  lbs/in^). 

Tension  and  compression  tests  were  not  made  in  connection  with  these  speci- 
mens. There  exist,  however,  tests  on  concrete  prisms  like  Fig.  21,  3  months 
old,  of  similar  composition,  of  which  the  average  of  three  strength  tests  were 
(72  =  8.8  kg/cm^  (125  lbs/in^),  and  0,1  =  1  j 2  kg/ cm-  (2446  lbs/in-),  so  that 


/=V  8.8X172=38.8  kg/cm2  (439  lbs/in2). 
The  average  of  the  three  shearing  tests  is, 
38.6  +  4'.7+3i-^' 


/=" 


=  37.1  kg/cm^  (528  lbs/in^). 


3.  Tests  with  reinforced  concrete  prisms. 
a.  With  straight  rods  only. 


36 


CONCRETE-STEEL  CONSTRUCTION 


The  experiments  were  performed  on  specimens  of  the  same  age,  size,  and 
mixture  as  the  foregoing;  but  each  specimen  was  reinforced  with  four  rods 
lo  mm.  (4/10  in.)  in  diameter,  near  the  upper  and  the  lower  surfaces,  as  illus- 
trated in  Fig.  37.  The  rods  were  not  connected  by  ties.  They  prevented  a 
rupture  of  the  specimen,  reduced  the  size  of  the  cracks,  and  allowed  the  load 
to  be  considerably  increased  after  one  shearing  crack  had  appeared  and  until, 
and  even  after,  the  other  crack  had  opened. 

r=i] 

Fig.  37. 

Specimen  i.  At  P  =  i2  t.  (26,400  lbs.)  a  fine,  low,  horizontal  crack  showed 
itself.  At.  P  =  iS  (33,000  lbs.)  a  fine  bending  crack  became  visible  in  the 
center,  and  shearing  took  place 

on  the  left  at    P  =  2o  t  (44,000  lbs.),  ^  =  31.0  kg/cm^  (441  lbs/in^), 
on  the  right  at  P  =  3o  /  (66,000  lbs.),  /=46.3  kg/cm^  (659  lbs/in^), 
Average,  /  =  38.6kg/cm2  (550  lbs/in^). 

In  spite  of  these  cracks,  the  load  was  increased  to  P=42  t.  (92,400  lbs.)  where 
the  sole  resistance  against  shear  was  the  sixteen  rod  sections  which  then  held 

/e  =  -^-°^— =  3350  kg/cm2  (47,650 lbs/in2). 

:6Xl2x- 
4 

Specimen  2  showed  shearing  cracks, 

on  the  left  at    P  =  i8  t  (39,600  lbs.),  1  =  2^.8  kg/cm^  (395  lbs/in^), 
on  the  right  at  F  =  2j  t  (59,400  lbs.),  /=4i.8  kg/cm^  (595  lbs/in^), 
Average,  ^  =  34-8  kg/cm^  (495  lbs/in^). 

The  load  was  increased  to  P  =  4o  t.  (88,000  lbs.)  at  which  point  a  horizontal 
crack  appeared  at  the  left  end.    For  this  load 

te  =  —  —  =3180  kg/cm2  (45,230  lbs/in2). 

i6X^-^Xi2 
4 

From  Table  I  on  page  17,  the  tensile  strength  of  the  reinforcement  can 
be  taken  as  4200  kg/cm^  (59,740  lbs/in^),  so  that  its  shearing  strength  would 
be  about  0.8X4200  =  3360  kg/cm^   (47,790  lbs/in^).    The  unequal  shearing 


SHEAR 


37 


resistances  on  the  left  and  right  can  be  explained  in  the  first  arrangement,  as 
due  to  an  unequal  distribution  of  the  load  P  on  the  two  plates.  In  the  latter 
case,  the  arithmetical  mean  gives  the  correct  value  of  the  shearing  strength. 

These  tests  show  that  the  shearing  cracks  appeared  in  the  reinforced  prisms 
at  practically  the  same  load  as  in  the  non-reinforced  ones,  and  conse(iuently  that 
only  after  the  shearing  strength  of  the  concrete  is  exceeded  does  that  of  the  iron 
come  into  play,  but  then  is  developed  to  its  full  value.  With  this  manner  of 
loading  for  pure  shear,  a  combination  of  the  strength  of  the  two  materials  thus 
seems  impossible  of  attainment.  In  any  case  final  rupture  depends  on  the 
resistance  of  the  steel. 

h.  With  some  bent  reinforcement. 

In  the  two  following  tests  (Fig.  38),  besides  two  straight  reinforcing  rods  10 
mm.  (4/10  in.)  in  diameter,  three  bent  ones  of  the  same  diameter  were  used, 


n 

^    — ^ 

QUO 


Fig.  38. 


and  so  designed  as  to  cut  the  shearing  planes  at  an  angle.    Otherwise,  the  size, 
shape,  and  mixture  were  as  before.    The  age  was  six  weeks. 
Specimen  i.  Shearing  crack, 

on  the  right  at  P  =  i8  t  (39,600  lbs.),  t  =  2j.S  kg/cm^  (395  lbs/in^), 
on  the  left  at    P=-3o  t  (66,000  lbs.),  1  =  46.4  kg/cm^  (660  lbs/in^), 
Average,  37.1  kg/cm^  (528  lbs/in^). 

The  load  was  increased  to  35  t.  (77,000  lbs.).  When  the  area  of  a  vertical 
section  through  the  bent  reinforcement  along  the  plane  of  shear  is  taken  into 
account,  the  area  is  increased  1.25  times,  and  the  unit  shearing  stress  is 


(4  +  6X1.25)- 


3870  kg/cm2  (55050  lbs/in2). 


Specimen  2.  Shearing  crack, 

on  the  left  at    P  =  i6  t  (35,200  lbs.),  ^  =  24.7  kg/cm^  (351  lbs/in^), 
on  the  right  at  ^  =  25  t  (55,000  lbs.),  /  =  38.7  kg/cm^  (551  lbs/in^). 
Average,       31.7  kg/cm^  (451  lbs/in^). 

The  load  was  increased  to  P  =  3o  t  (66,000  lbs.);  /e  =  33io  kg/cm^  (47,080 
lbs/in2). 


38  CONCRETE-STEEL  CONSTRUCTION 

Specimen  3.  A  bending  crack  appeared  at  P  =  i2  t  (26,400  lbs.) ;  shearing 
occurred 

at  the  left  at    P  =  i5  t  (33,000  ibs.),  /  =  23.2  kg/cm^  (330  lbs/in^), 
at  the  right  at  ^  =  28  t  (61,600  lbs.),  /  =  43.3  kg/cm^  (616  lbs/in^), 
Average,       33.3  kg/cm2  (473  lbs/in^). 


Fig.  39. 


Solid  Cylinders. 


Fig.  40. 


Fig.  41. 


Fig. 


Hollow  Cylinders. 


The  load  was  increased  to  P  =  32  t  (70,400  lbs.);  ^e  =  3540  l^g/cm2  (50,350 
lbs/in2). 

Consequently,  the  same  observations  apply  to  tests  b  as  to  tests  a. 


TORSION 


39 


Torsion  Experiments  with  Concrete  Cylinders. — In  a  cylinder  under- 
going a  twist,  without  any  axial  forces  at  i)lay,  no  normal  stresses  exist  within 
any  section,  only  shearing  stresses  acting,  and  at  each  point  the  latter  are  equal 
along  directions  parallel  and  perj)endicular  to  the  axis,  so  that  all  elements  in 
the  body  are  stressed,  as  is  illustrated  in  Fig.  31,  page  32. 

It  has  been  shown  by  the  shearing  experiments  that  the  resistance  offered 
by  concrete  to  shear  is  somewhat  greater  than  its  tensile  strength.  Consequently 
rupture  of  a  cylinder  subject  to  torsion  must  take  place  along  a  screw  surface 
with  a  i)itch  of  45°  at  right  angles  to  the  major  dilation  or  the  oblic^ue  tensile 
stresses.    (See  Figs.  39-42.) 

These  torsion  experiments  were  made  at  the  Testing  Laboratory  of  the  Royal 
Technical  High  School  of  Stuttgart.  The  mixture  of  the  concrete  was  1:4,  and 
its  age  2  to  3  months. 

a.  Solid  cylinder,  26  cm.  (10.24  in.)  in  diameter.  The  length  of  the  speci- 
men under  test  was  34  cm.  (13.38  in.).  (See  Figs.  39  and  40.)  The  twisting 
moment  was  applied  on  the  hexagonal  heads.    (See  Table  XIII.) 


Table  XIII 

TORSIONAL  STRENGTH  OF  SOLID  CYLINDERS 


No. 

Torque  M^i 

Torsional  Strength  according  to 

the  Formula  "<=  ■ 

16 

Age  in  Days. 

kg/cm 

in  /lbs 

kg/ cm^ 

lb/in2 

V 
\T 
VII 
VIII 

01 500 

66500 
46000 
59500 

53300 
57600 
39800 
51500 

19-3 
13-3 
17.6 

259 
275 
189 
250 

89 
85 
79 
98 

Average. . .  |         17.  i 

243 

b.  Hollow  cylinders  of  the  same  external  dimensions,  with  inner  diameters 
about  do  =  is  cm.  (5.9  in.),  gave  the  torsion  moments  shown  in  Table  XIV. 


Table  XIV 

TORSIONAL  STRENGTH  OF  HOLLOW  CYLINDERS 


No. 

Torque  Mj. 

Torsional  Strength, 

Age  in  Days. 

kg/cm 

in/lb 

kg/cm^ 

ibs/in2 

XVI 
XVII 
XVIII 

30000 
24500 
29000 

26000 
21  200 
25100 

9-4 
7-9 
9-3 

134 
112 
132 

54 
55 
52 

Average .... 

27830 

24100 

8-9 

126 

40 


CONCRETE-STEEL  CONSTRUCTION 


The  tensile  strength  of  some  hollow  cylinders  of  similar  section  and  equal 
age  provided  with  the  corresponding  heads,  gave  an  average  of  ^72  =  8.0  kg/cm^ 
(113.8  lbs/in^),  while  the  similar  above  described  tensile  specimens,  like  Fig.  21, 
gave  7.7  kg/cm^  (109.5  lbs/in^).  The  results  found  from  the  hollow  cylinders 
agree  quite  satisfactorily  with  each  other,  while  the  above  described  theory  for 
solid  cylinders  has  not  been  confirmed. 

Aside  from  the  greater  age  of  the  solid  cylinders,  the  greater  value  of  Td  is 
explained  on  the  ground  that  since  the  modulus  of  elasticity  diminishes  with 
increase  of  stress,  the  sections  near  the  center  carry  a  relatively  large  part  of 
the  load,  as  is  shown  by  the  formula 

16 

so  that  the  load  is  reduced  on  the  outer  portion.  The  torsional  strength  of  con- 
crete, therefore,  bears  the  same  relation  to  its  tensile  strength  as  do  the  bend- 
ing and  tensile  strengths.  In  this  manner  can  be  explained  the  high  value  of 
1 7.1  kg/cm^  (243  lbs/in^),  when  compared  with  the  tension  test  specimens  of 
the  same  material  and  mixture  which  gave  about  9  kg/cm^  (128  lbs/in^),  when 
3  months  old.  And  with  hollow  cylinders,  in  which  the  rupture  takes  place 
along  a  screw  surface  with  a  45°  pitch  and  at  right  angles  to  the  maximum  tensile 
strains,  the  computed  torsional  stresses  also  correspond  with  the  actual  ones. 
It  must  be  mentioned,  however,  that  only  through  the  use  of  extremely  plastic 
concrete  will  this  agreement  be  obtained,  and  only  with  wet  concrete  can  the 
tamping  be  thoroughly  effective,  as  is  especially  necessary  with  hollow  cylinders. 

With  regard  to  torsion  investigations  concerning  spirally  reinforced  concrete 
hollow  cylinders,  see  page  53,  "  The  extensibility  of  concrete." 

Shearing  Experiments  with  Slotted  Concrete  Beams. — These  tests 
were  conducted  on  specimens  with  slits  molded  along  the  neutral  axis,  so  that 
with  the  method  of  loading  shown  in  Fig.  43,  the  failure  would  take  place  by 


Fig.  43. 


a  shearing  of  the  connecting  bridges  at  the  ends.  The  tests  were  made  at  the 
Testing  Laboratory  at  Stuttgart. 

At  the  ultimate  load,  the  shearing  stresses  existing  in  the  sections  a-a,  are 
calculated  as  follows: 


TORSION 


41 


The  unit  shear  at  any  point  x  along  the  neutral  plane  is  * 

P  S 


2  Jb' 


where  S  is  the  statical  moment  of  the  cross-section  lying  above  the  neutral  axis 
in  relation  to  it,  and  /  is  the  moment  of  inertia  of  the  whole  section.    Thus  the 


total  shear  from  o  to  —  is 

2 


r-r^)-  =  -  —  - 

2       2    J  2 


ana  the  shearing  strength  in  a-a  is  given  by 

_P  S  I 

It  must  be  explained  that  the  side  subject  to  tensile  stresses  had  to  be  rein- 
forced, so  that  the  weakest  points  in  the  body  would  be  the  bridges  over  the 
supports,  and  so  that  the  specimen  would  not  fail  prematurely  through  tension. 

 ^  .  Al  .  . 


1  \ay- 

37$ 

V 

-  -  .  ^.  .m...-^.-..-  

 3.75_  

 1^ 

30 

— — 1^  — 

30 



A 

A       ■  ■ 

1 

1  1 

 -t  — 

Fig.  44. — Slotted  beam  shear  test. 

Further,  the  bodies  were  not  supported  accurately  under  the  centers  of  the  bridges, 
so  that  some  bending  was  experienced  at  those  points.  This  would  produce 
a  result  equivalent  to  a  partial  reduction  in  the  effective  width  of  the  bridges, 
as  compared  with  the  original  arrangement. 

Example.    Specimen  "85,"  wet  mixture,  1:3,  age  105  days. 

Under  a  load  P  =  i43o  kg.  (3146  lbs.),  the  crack  hi  appeared  conspicuously 
through  the  whole  bridge. 

At  P  =  i620  kg.  (3564  lbs.),  a\  showed  itself  through  the  whole  bridge,  and 
m\  started  in  the  edge. 

J'^y                       pl            My  C^My 
pbdy,  but  M  =  —  or  p  =  ,  so  that  stress  =  I   bdy  = 
o                        y             I                           Jo  I 

M  ry  r 

—  j    bdy.    M=  j  Fdx=Fdx  when  x  is  very  small,  so  that  F  means  stress  in  differential 

Fdx  fy 

length  of  beam.    Total  stress=          I    hdy.    Divide  by  area  over  which  total  stress  acts  = 

^  Jo 

Fdx  fy  F 

bdx  to  get  unit  stress.    Unit  stress  =          I  bdy= — S,  where  5  is  static  moment  of  section 

Ibdx J  lb 

above  neutral  axis  about  that  axis. — Trans. 


42 


CONCRETE-STEEL  CONSTRUCTION 


At  P  =  i77o  kg.  (3894  lbs.),  a2  appeared. 
At  P  =  20oo  kg.  (4400  lbs.),  W2  appeared. 

Under  a  load  of  P  =  24io  kg.  (5302  lbs.),  a  wide  crack  formed  at  W3  and 
W2  widened  considerably.    The  load  could  not  be  further  increased. 

In  Table  XV  the  observed  shearing  strengths  are  given,  together  with 
the  tensile  and  compressive  strengths  of  the  specimens  illustrated  in  Fig.  21 
(page  21).    The  results  are,  each,  averages  of  three  specimens. 


Table  XV 

SHEARING,  TENSILE  AND  COMPRESSING  UNIT  STRENGTHS 


Mixture. 

I 

3 

I 

4 

1 :  7 

Per  cent  Water 
Unit  Stress. 

8% 

14% 

8% 

14% 

8% 

14 

% 

e 

\ 

B 

a: 

"c 
■<, 

w 

B 
0 

s 

"s 

\ 

B 
0 

B 

Shear  

512 

30 

427 

31 

441 

28 

398 

26 

370 

19 

270 

Tension  

12.6 

179 

149 

9-2 

131 

8.8 

125 

4.4 

63 

5-5 

78 

280 

398 

T95 

2770 

220 

3130 

153 

2180 

127 

I8I0 

88 

125  I 

The  shearing  strength  for  1:4,  here  observed,  of  from  31  to  28  kg/cm^  (441 
to  398  lbs/in^),  is  a  little  smaller  than  the  one  found  by  direct  shear,  of  37  kg/cm^ 
(526  lbs/in^).  The  reason  probably  lies  in  the  not  entirely  rigorous  methods 
of  calculation  used  in  connection  with  the  slotted  prisms,  or  else  in  that  the  solid 
end  connections  had  an  appreciable  thickness,  so  that  partially  inclined  cracks 
could  occur  from  diagonal  tension. 

In  practice,  the  case  of  pure  shear  is  very  rare.  Diagonal  tensile  stresses 
are  always  combined  with  shearing  ones,  and  the  former  become  of  critical  impor- 
tance long  before  the  shear  does,  as  the  torsion  experiments  plainly  show.  Th's 
point  will  later  be  discussed  more  fully,  in  connection  with  shearing  tests  on 
beams. 

Adhesion  or  Sliding  Resistance  between  Steel  and  Concrete.  Experi- 
ments aiming  at  the  determination  of  the  amount  of  adhesion  between  concrete 
and  embedded  steel,  or  the  resistance  offered  to  sliding,  can  be  carried  out  in 
various  ways.  The  resistance  experienced  by  an  embedded  rod  when  drawn 
out  can  be  directly  measured,  or  the  adhesion  may  be  ascertained  by  computation 
from  bending  tests.  Consideration  will  later  be  given  to  a  discussion  of  experi- 
ments of  the  latter  kind  concerning  adhesion,  which  naturally  are  of  the  greatest 
importance  in  this  subject. 

The  published  figures  for  directly  ascertained  adhesion,  disclose  many  dif- 
ferences, produced  by  the  variableness  of  concrete,  the  method  of  test,  the  nature 
of  the  surface  of  the  rod,  etc.  For  a  long  period  the  value  of  the  unit  adhesion 
of  from  40  to  47  kg/cm^  (569  to  668  lbs/in^),  determined  by  Bauschinger  in  his 
nivestigations  for  the  A.-G.  fiir  Beton  &  Monierbau,  was  accepted.  Among 
later  experiments  may  be  mentioned  those  of  Tedesco,*  with  six-day  old  mortar 

*  "Du  Calcul  des  ouvrages  en  ciment  avec  ossature  metallique,"  by  MM.  Ed.  Coignet  and  N. 
De  Tedesco,  Paris,  1894. 


ADHESION 


43 


i  IG.  45. 


prisms  which  gave  an  adhesion  of  20  to  25  kg/cm^  (284  to  355  lbs/in^),  and  those 
of  the  "  Service  franjaise  des  phares  et  balises,"  *  with  25  to  36  mm.  (i  to 
1 1  in.  approximately)  round  rods  which  were  anchored 
;  for  a  length  of  60  cm.  (23.6  in.)  with  Portland  cement 

into  stone  blocks.    After  setting  for  a  month  in  the  open 
f     air,  the  rods  were  pulled  out.     The  adhesion  \\as  found 
;     to  vary  with  the  diameter  of  the  rod,  and  was  between  20 
\     and  48  kg/ cm-  (284  and  682  lbs/in^), 
j  The  larger  values  correspond  with  thicker  rods,  and 

i     with  material  possessing  a  higher  elastic  limit.  With 
equal  sections,  the  stress  was  quite  constant,  and  was 
practically  equal  to  the  elastic  limit  of  the  steel  involved. 
Thus,   the  adhesion  between  concrete  and  steel  was 
broken  when  the  sections  of  the  rods  began  to  diminish  perceptibly. 

With  a  slow,  regular  withdrawal  of  the  rods,  almost  as  large  a  sliding  resis- 
tance  was   disclosed,  which  varied   between  39  and  71 
kg/cm-  (555  and  loio  lbs/in-)   of  surface  of  contact.  { 
The  variation  in  this  sliding  resistance  may  be  explained 
by  the  fact  that  the  surface  of  commercial  rod  iron  is 
not  a  mathematical  cylinder. 

Some  experiments  were  made  by  the  writer  in  1904 
on  the  "  pressing  through  "  of  rods  set  in  concrete,  as  is 
illustrated  in  Fig.  45.  The  cubes  were  20  cm.  (7.8  in.) 
on  an  edge;  the  concrete  was  mixed  in  the  proportions 
of  I  to  4,  with  different  percentages  of  water,  and  was 
four  weeks  old.  The  specimens  did  not  crack,  and 
it  was  shown  that  after  the  adhesion  was  overcome, 
there  existed  a  considerable  constant  sliding  resistance. 

A  second  series  of  tests  was  made  upon  exactly  similar 
cubes  with  20  mm.  (|  in.  approx.)  rods,  and  special 
precautions  were  taken  to  prevent  cracking  of  the  con- 
crete by  embedding  in  it  a  4.5  mm.  (3/16  in.  approx.) 
wire  spiral  with  3  cm.  (1.2  in.)  pitch  and  10  cm,  (3.9 
in.)  diameter.  (Fig.  46.)  The  age  of  the  specimens  was  four  weeks.  The 
results  are  as  given  in  Table  XVL 


Table  XVI 
ADHESION  TO  ROUND  RODS 


Per  Cent  of  Water. 

Adhesion  from  Average  of  Four  Specimens. 

Without  Spiral. 

With  Spiral. 

kg  'cm2 

lbs /in  2 

kg  cm 2 

lbs  in  2 

10 

48.8 

694 

50.8 

723 

12.5 

31.2 

444 

45-9 

653 

15 

29. 1 

414 

54-0 

768 

*  Annales  des  ponts  et  chaussees,  1898,  III. 


44 


CONCRETE-STEEL  CONSTRUCTION 


The  percentage  of  water  given  is  only  nominal,  since  the  sand  and  gravel  were 
moist.  The  pressure  of  the  testing  machine  was  increased  rather  rapidly  for 
the  larger  loads. 

The  results  approach  closely  the  shearing  strength  of  similar  concrete  speci- 
mens. The  compressive  stresses  in  the  rods  reached  a  maximum  of  2140  kg/cm^ 
(30,440  lbs/in^),  and  consequently  were  below  their  observed  elastic  limit  of 
from  2600  to  3200  kg/cm^  (36,980  to  45,520  lbs/in^). 

Although  the  non-reinforced  concrete  cubes  were  not  cracked  by  the  pressing 
through  of  the  rods,  their  adhesive  strength  was  smaller  than  was  that  of  the 
ones  containing  spirals. 

The  results  of  some  American  tests  were  pubHshed  in  Engineering  News," 
1904,  No.  10.  The  adhesive  strength  of  rods  of  different  shapes  was  examined 
for  a  mortar  mix  of  i  to  3  and  for  various  concrete  mixtures.  With  cubes  of 
cement  mortar,  15  cm.  (6  ins.)  on  an  edge,  the  average  values  of  Table  XVII 
were  obtained: 


Table  XVII 

AMERICAN  ADHESION  TESTS 


Section. 

Unit  Steel  Stress 
Oe 

Adhesive  Strength. 

kg/cm^ 

lbs/in2 

kg/cm^ 

lbs/in2 

Square, 

1570 

22330 

30.2 

430 

Round, 

1780 

25320 

35-8 

509 

Flat, 

1270 

18060 

20.5 

292 

Square, 

2430 

34560 

25.8 

364 

It  is  seen  from  this  that  the  round  rods  developed  a  greater  adhesive  strength 
than  the  square  ones,  and  considerably  more  than  the  flat  iron. 

Some  concrete  prisms  20  by  20  cm.  (7.8  in.)  in  section  and  25  cm.  (10  in.) 
high,  contained  square  rods  25  by  25  mm.  (i  in.  approx.),  and  developed  adhe- 
sive strengths  of  34  to  41  kg/cm^  (484  to  583  lbs/in^),  or  an  average  of  37.5 
kg/cm^  (533  lbs/in^),  which  agrees  well  with  the  values  found  by  Europeans. 

In  a  very  careful  and  exhaustive  manner.  Bach  carried  out  a  series  of 
tests  on  the  sliding  resistance  of  steel  embedded  in  concrete,*  for  the  investiga- 
tions of  the  Eisenbetonausschusses  der  Jubilaumsstiftung  der  Deutschen  Indus- 
trie.   His  results  shed  new  light  on  the  subject  of  adhesion. 

The  concrete  specimens  were  made  in  the  form  of  square  prisms  22  cm. 
(8.7  ins.)  on  a  side  and  with  heights  of  10,  15,  20,  25,  and  30  cm.  (4,  6,  8,  10, 
12  in.  approx.).  The  concrete  was  mixed  in  the  proportions  of  1:4,  with  Rhine 
sand  and  gravel,  of  which  the  aggregate  contained  3  parts  sand  of  o  to  5  mm. 
(o  to  0.2  in.)  grains  and  2  parts  gravel  of  5  to  15  mm.  (0.2  to  0.6  in.)  pebbles. 
Heidelberg  Portland  cement  was  used,  so  that  the  specimens  were  exactly  like 
those  described  on  pages  43  and  44. 

*"Versuche  iiber  den  Gleitwiderstand  einbetonierter  Eisen,"  by  C.  v.  Bach,  Berlin,  1905, 
and  also  No.  22  of  the  "Mitteilungen  iiber  Forschungsarbeiten  auf  dem  Gebiete  des  Ingenieur- 
wesens." 


ADHESION 


45 


The  experiments  included  tests  for  the  determination  of  the  influence  of 
the  amount  of  water  used,  the  quantity  of  sand,  the  influence  of  jarring  the 
specimen  before  the  concrete  had  set,  and  finally,  time  tests  of  specimens  up  to 
three  months  old.    The  following  conclusions  were  deduced: 

That  percentage  of  water  was  best  with  which  it  was  just  possible  to  manu- 
facture the  specimens  satisfactorily.  With  the  proportions  above  described, 
this  was  12  per  cent. 

Within  certain  limits,  the  relative  proportions  of  sand  and  gravel  have  no 
important  influence  on  the  resistance  to  sliding,  so  long  as  the  percentage  of 
water  is  proportionately  small  when  small  amounts  of  sand  are  used. 

The  resistance  to  sliding  will  be  increased  by  jarring  the  finished  specimen 
before  setting  is  completed,  at  least  when  the  specimen  stands  on  a  wooden 
bottom,  which  gets  jarred  by  being  struck  by  other  bodies.  This  increase  is 
more  important  when  small  percentages  of  water  are  used,  and  is  to  be  explained 
by  the  fact  that,  through  the  jarring,  the  grout  which  is  necessary  to  a  good 
bond  will  be  enabled  to  collect  around  the  reinforcement. 

The  sliding  resistance  is  considerably  greater  in  tests  conducted  at  high 
rates  of  speed  than  at  slower  ones  where  the  loads  act  for  longer  periods  at  each 
step.  Also,  tests  in  which  rods  are  "  pushed  through  "  are  somewhat  higher 
than  when  they  are     pulled  through." 

In  regard  to  the  practical  employment  of  these  results,  it  is  to  be  noted  pri- 
marily that  it  is  impossible  to  obtain,  in  actual  work,  the  exact  percentage  of 
water  above  mentioned,  on  account  of  humidity  of  the  various  aggregates,  but 
that  it  is  necessary  to  rely  almost  entirely  on  experience  and  good  practice.  On 
the  other  hand,  an  excess  of  water  does  not  then  have  the  harmful  effect  that 
it  does  on  test  specimens  molded  in  solid  cast-iron  forms,  since  the  wooden  molds 
absorb  a  part  of  the  water  and  some  more  is  lost  through  the  cracks  between 
the  boards.  Furthermore,  in  building  construction,  the  fresh  concrete  will 
receive  plenty  of  jarring  from  the  forms,  so  that  the  highest  value  obtained  from 
the  experiments,  in  which  the  specimens  were  shaken  as  well  as  tamped,  may 
be  assumed  as  a  proper  working  stress. 

A  very  important  point,  and  one  here  brought  out  for  the  first  time,  is  that 
for  steel  stresses  far  below  the  elastic  limit,  the  unit  adhesion  diminishes  with 
the  length  of  rod  embedded.  The  explanation  of  this  phenomena  is  as  follows: 
The  tensile  stress  in  the  rod  will  decrease  from  the  outside  of  the  concrete  to 
the  inner  end  of  the  rod  as  the  stress  is  transferred  from  its  surface  to  the  con- 
crete. Because  of  its  elasticity,  the  rod  will  stretch  under  the  tension,  while 
the  concrete  will  be  throw^n  into  compression  and  will  shorten.  Consequently, 
even  under  small  tensile  stress,  because  of  the  changes  of  length  in  opposite 
directions  in  the  two  materials,  a  sliding  effect  will  be  produced  along  the  rod, 
near  its  outer  end,  so  that  the  tensile  stress  in  the  steel  will  not  be  uniformly 
distributed  over  the  whole  length  of  the  rod  embedded  in  the  concrete.  It  will 
first  be  taken  up  by  the  adhesion  at  the  outer  end,  and  only  after  that  is  exceeded 
and  a  slight  displacement  takes  place,  will  the  distant  parts  of  the  concrete  be 
stressed.  It  follows  from  this  unequal  distribution  of  stress,  that  the  observed 
values  of  this  stress  are  too  small  and  that  they  should  more  properly  be  termed 
the  "  frictional  resistance,"  as  Bach  has  done.    The  shorter  is  the  embedded 


46 


CONCRETE-STEEL  CONSTRUCTION 


length  of  rod,  the  smaller  are  the  tension  and  elongation,  and  the  more  nearly 
equally  distributed  will  be  the  effect  over  the  whole  surface. 

When  the  rods  are  pushed  through,  there  exist  practically  the  same  con- 
ditions, but  in  less  degree,  because  then  the  steel  and  concrete  are  loaded  in  like 
kind.  Even  then  a  slight  sliding  will  occur  very  early  along  the  outer  portions 
of  the  rod.  This  slight  sliding  explains  the  influence  shown  by  the  rate  of  appli- 
cation of  the  load.  It  is  easily  seen  that  with  a  high  rate,  the  sliding  does  not 
have  time  to  develop,  and  that  the  adhesive  stress  is  then  more  uniformly  dis- 
tributed over  the  embedded  area  of  the  rod. 

In  Fig.  47  are  given  the  principal  results  of  Bach's  tests.  They  refer  entirely 
to  1:4  concrete  prisms  with  15  per  cent  of  water.    The  earlier  tests  were  con- 


jI,  o  fOO        f^if        ^00        Jl^        500  >n^ 

Fig.  4/. — Results  of  adhesion  experiments. 

ducted  with  applications  of  load  for  short  periods — each  step  occupied  one-half 
a  minute  (which  is  really  long  as  compared  with  most  experiments).  The 
embedded  lengths  of  the  rods  are  plotted  as  abscissas,  and  the  observed  resis- 
tances to  sliding,  as  ordinates.  If  the  curves  for  adhesion  on  pushed  and  pulled 
rods  under  short  load  periods,  and  also  the  curve  shewing  the  results  for  longer 
duration  of  load  (from  nothing  up  to  no  minutes)  are  extended  to  intersect  the 
axis  of  ordinates,  all  three  meet  at  practically  the  same  point,  which  corresponds 
with  an  adhesive  strength  of  38  kg/ cm-  (540  lbs/in^). 

At  this  value,  which  corresponds  to  a  length  /  =  o,  the  influences  of  the  embedded 
length  of  rod,  of  premature  sliding,  of  time,  and  the  difference  between  pulling 
and  pushing,  all  vanish.  This  value  of  38  kg/cm^  (540  lbs/in^),  happens  to 
correspond  with  that  found  by  the  author  on  specimens  of  the  same  mixture  and 
age  for  shearing  strength,  and  also  approaches  closely  that  for  the  quickly  oper- 
ated adhesion  experiments.  The  low  point  of  the  middle  curve  at  /  =  2oo  mm. 
may  be  explained  by  the  fact  that  those  specimens  were  first  manufactured,  and 
the  operator  had  not  yet  acquired  proper  experience. 

In  addition  to  the  experiments  on  the  slipping  resistance  of  embedded  round 
Tods,  made  at  the  Testing  Laboratory  in  Stuttgart,  a  series  was  also  conducted 
with  Thacher  *  bars.    The  specimens  were  again  prepared  of  the  same  mixture 

*  Versuche  mit  einbetonierten  Thatchereisen  von  D.-Ing.,  C.  v.  Bach,  Berlin,  1907. 


ADHESION 


47 


of  I  part  of  Portland  cement  and  4  parts  sand  and  gravel,  with  15  per  cent  water. 
The  height  of  the  specimens  was  20  cm.  (7.9  in.),  while  the  length  of  the  side 
of  the  square  base  was,  in  some  cases,  22  cm.  (8.7  in.),  some  16  cm.  (6.3  in.), 
and  some  10  cm.  (3.9  in.),  and  the  resistance  to  pulling  out  was  found  to  vary 
with  the  diameter  of  the  specimen,  since  all  split  when  the  Thacher  rods  were 
withdrawn.  If  the  pull  P  is  uniformly  distributed  over  the  embedded  surface  (O), 
the  resistance  to  sliding  for  the  several  specimens  was  as  given  in  Table  XVIII. 


Table  XVIII 

UNIT  ADHESION  FOR  DIFFERENT  LENGTHS  OF  EMBEDMENT 


Metric. 

English. 

Metric. 

English. 

Metric. 

English. 

Length  of  side  

^*max. 

22  cm. 
58-5 

8.7  in. 

832 

16  cm. 
56-1 

6.3  in. 

799 

10  cm. 
33-4 

3-9  in- 

475 

0 

It  is  evident  from  the  last  figure  that  with  a  minimum  thickness  of  specimen 
equal  to  3.75  cm.  (1.5  in.),  and  with  lesser  values,  the  splitting  effect  of  the  knots 


Fig.  48. 


is  so  great  that  greater  adhesion  cannot  be  expected  than  that  of  common  round 
rods  as  they  come  from  the  mills. 

With  greater  thickness  of  concrete  the  splitting  occurred  when  the  elastic 
limit  of  the  steel  had  been  reached. 

Only  those  adhesion  experiments  in  which  the  steel  stress  remains  under 
the  elastic  limit,  give  a  proper  value  of  the  adhesive  strength  to  be  used  in  the 
design  of  reinforced  concrete  structures,  and  consequendy  all  steel  must  be  so 
arranged  as  to  length,  shape,  and  thickness  that  it  will  effect  a  safe  transfer  of 
stress  to  the  concrete.    Actual  tensile  stresses  are  usually  small,  however,  and 


48 


CONCRETE-STEEL  CONSTRUCTION 


an  increase  up  to  the  elastic  limit  through  overloading  of  beams  is  seldom  to 
be  feared. 

In  singly  reinforced  slabs,  the  ends  of  the  rods  rest  in  large  masses  of  concrete, 
so  that  a  diminishing  of  the  adhesion,  because  of  premature  cracking  of  the 
surrounding  concrete,  is  not  to  be  feared.  In  slabs,  the  amount  of  the  embedding 
is  less,  but  near  the  ends  of  beams,  stirrups  are  introduced  which  surround  the 
concrete  to  some  extent,  and  so  preserve  its  adhesive  strength.  In  this  connec- 
tion are  here  given  the  valuable  results  of  the  French  Reinforced  Concrete  Com- 
mission's *  experiments:  Certain  prisms  with  centrally  located  rods  were  manu- 
factured, in  which  only  2  to  2.5  cm.  (0.8  so  i.o  in.)  of  concrete  existed  between  the  rod 
and  the  outside  surface.  Besides  these,  some  were  made  without  stirrups,  as 
illustrated  in  Fig.  49.  A  second  series  had  three  flat  iron  stirrups  30  by  2  mm. 
(lA  by  ins.)  as  are  used  in  the  Hennebique  system,  and  which  enclosed 
the  30  mm.  (ifg  in.)  diameter  rods  tightly.  In  the  third  series,  open  stirrups 
of  the  same  flat  iron  were  employed,  which  enclosed  a  larger  mass  of  concrete 
and  were  separated  from  the  rods  by  a  space  of  about  i  cm.  (|  in.).  The 
concrete  was  composed  of  300  kg.  (661  lbs.)  of  cement,  400  1  (14  cu.ft.)  sand, 
and  800  1  (28  cu.ft.)  gravel,  with  8.8  per  cent  by  weight  of  water,  and  were  six 
months  old  at  the  time  of  the  test.  The  resistances  shown  in  Table  XIX  were 
developed  against  pulling  the  rods  out  of  the  concrete: 


Table  XIX 

ADHESION  IN  THE  PRESENCE  OF  STIRRUPS 


Specimen. 

Starting  Resistance. 

Average  Sliding  Resistance. 

Stirrups. 

Figure. 

No.  of 

kg/cm2  of 

kg /cm-  of 

Specimens. 

Surface. 

lbs/in  2 

Surface. 

lbs/in2 

None  

49a 

2 

■  7-2 

102 

8.1 

115 

19.9 

283 

14.2 

202 

Hennebique.. 

49& 

2 

20.0 

284 

17.2 

245 

16.9 

240 

12.8 

182 

Open  

49c 

2 

25-7 

366 

18.2 

259 

29.8 

424 

21.2 

302 

A  repetition  of  these  experiments  with  specimens  three  months  old  (in  which 
the  stirrups  of  flat  iron  were  replaced  with  9  mm.  (JJ  in.)  rods,  gave  higher 
results,  (see  Table  XX),  each  being  the  average  of  three  specimens: 


Table  XX 

ADHESION  IN  THE  PRESENCE  OF  STIRRUPS 


Adhesion. 

Sliding  Resistance. 

Specimen. 

kg/cm^ 

lbs/in2 

kg/cm  2 

lbs/in2 

Figure  49a 

24-7 

8.8 

125 

Figure  496 

26. 1 

371 

17.7 

252 

Figure  49c 

31.2 

457 

30.0 

284 

*  Commission  du  ciment  arme.  Experiences,  rapports,  etc.,  relatives  a  I'emploi  du  beton 
arme.    Paris,  1907. 


ADHESION 


49 


It  is  seen  that  the  open  stirrups  which  surround  the  concrete  have  an  advan- 
tage over  those  which  enclose  the  rods  more  tightly,  and  that  the  adhesion  then 
developed  corresponds  well  with  the  Stuttgart  results  with  rods  in  the  centers 
of  the  specimens.  Of  interest  are  also  the  adhesion  experiments  of  the  French 
Commission  on  an  old  reinforced  concrete  beam,  in  which  5  mm.  in.)  steel 
wire,  because  of  its  somewhat  crooked  nature,  developed  a  resistance  to  sliding 
of  80  to  92  kg/ cm-  (1138  to  1308  lbs/ in-).  These  results  are  of  practical  impor- 
tance, since  small  rods  are  never  absolutely  straight. 


CHAPTER  IV 


THEORY  OF  REINFORCED  CONCRETE 

EXTENSIBILITY 

Extensibility  of  Reinforced  Concrete. — Experiments  with  straight  rein- 
forced concrete  prisms,  in  which  the  extension  is  produced  by  axial  tensile 
stresses,  have  several  disadvantages.  These  consist  primarily  in  the  great  trouble 
in  securing  an  exactly  central  application  of  the  tensile  stress,  and  the  fact  that 
the  force  can  be  transferred  to  the  reinforcement  only  through  large  adhesive 
stresses,  so  that  the  ends  of  the  specimens  crack  prematurely.  In  the  first  edition 
of  this  book  were  given  several  theoretical  investigations  concerning  combina- 
tions of  steel  and  concrete  under  certain  assumed  conditions  with  regard  to 
the  elasticities  of  the  two  materials. 

Of  much  more  value  are  tests  in  which  the  extensibility  of  reinforced  concrete 
is  ascertained  through  experiments  on  specimens  subjected  to  flexure.  Of 
such,  the  best  known  are  Considere's.  His  first  tests  *  were  made  on  mortar 
prisms  of  square  section  6  cm.  (2.4  ins.)  on  a  side,  and  60  cm.  (23.6  ins.)  high, 
reinforced  on  the  stretched  side  by  round  steel  rods.  The  prisms  were  tested 
by  fixing  one  end  and  applying  at  the  other  a  bending  moment  in  such  manner 
that  it  w^as  constant  for  all  sections.  The  extension  of  the  stretched  side  was 
then  measured  with  each  increase  cf  load.  The  mixture  was  1:3,  and  the  rein- 
forcement consisted  of  three  round  rods  4.25  mm.  in.  approx.)  in  diameter. 
For  comparison,  a  few  prisms  had  no  reinforcement.  With  one  prism,  the 
bending  moment  w^as  increased  so  that  the  tension  side  was  stretched  2  mm. 
per  meter  (0.002  ft.  per  foot).  Then  a  moment  was  applied  139,000  times, 
which  was  from  44  to  71  per  cent  of  the  first  moment,  and  after  each  application 
the  return  to  the  initial  condition  was  complete.  These  repeated  applications 
gave  extensions  of  from  0.545  to  1.25  mm.  per  m.  (0.000545  to  0.00125 
foot).  Small  strips  12  by  15  mm.  (0.5  by  0.6  in.)  in  section  were  then  cut  from 
the  prisms,  and  the  bending  moment  again  api)lied.  The  resulting  strength 
was  surprisingly  high,  almost  equal  to  that  of  fresh  specimens.  From  the  com- 
parative tests  of  non-reinforced  mortar  prisms,  it  was  found  that  the  ultimate 
flexure  was  between  o.i  and  0.2  mm.  (0.04  and  0.08  in.).  It  thus  follow:"  that 
in  a  reinforced  concrete  body  the  reinforcement  gives  the  concrete  the  ability 
to  bend  to  a  considerably  greater  extent  than  when  plain. 

Considere  explains  this  as  follows:  As  is  known,  in  a  metal  rod  subjected 
to  tensile  stress,  the  latter  is  at  first  distributed  uniformly  throughout  the  whole 
length;  but  with  increase  of  stress  the  rod  contracts  at  some  point,  and  will 

*  Genie  Civil,  1899. 

60 


EXTENSIBILITY 


51 


then  undergo  considerable  stretching.  Thus,  the  total  measured  length  may 
have  increased  only  20  per  cent,  while  in  the  neighborhood  of  the  point  of  ru))ture 
the  actual  stretch  has  been  10  to  15  times  this  amount.  If  it  is  suj)])c)S'j(l  that 
the  ])henomenon  known  as  redaction  in  area  "  also  applies  to  cement  mortar, 
then  the  total  elongation  measured  between  the  ends  will  give  only  an  average 
value,  and  the  mortar  will,  in  reality,  possess  a  very  much  greater  ability  to 
stretch  than  this  value  represents.  In  reinforced  construction  the  concrete  is 
attached  to  the  steel,  which  latter  possesses  a  much  higher  elastic  limit  than 
does  the  concrete.  When  undergoing  stress,  therefore,  the  steel  will  still  tend 
to  have  the  extension  distributed  uniform.ly  over  its  whole  length,  at  a  stress  at 
which  the  concrete  tends  to  contract  locally.  But  the  adhesion  makes  it  neces- 
sary for  the  concrete  to  follow  the  steel  in  its  extensibility.  It  will  therefore 
endure  throughout  its  whole  length  the  maximum  possible  deformation,  and 
rupture  will  finally  take  place  with  an  elongation  (measured  over  all)  which  is 
considerably  larger  than  if  reinforcement  were  present.  This  explanation  given 
by  Considere  is  obvious,  if  the  phenomenon  of  "  reduction  of  area  "  really  exists 
in  concrete. 

In  computations  concerning  these  bending  tests,  Considere  employed  a 
method  with  reference  to  the  relative  distribution  of  stress  between  the  concrete 
and  steel,  which  made  the  concrete  show  no  greater  tensile  strength  than  that 
developed  by  plain  concrete  prisms.  This  method  was  not  entirely  free  from 
objections,  and  therefore  Considlre  subsequently  made  some  true  tension  tests 
with  reinforced  concrete  prisms.*  Mortar  prisms  of  square  section,  47  mm. 
(1.85  ins.)  on  a  side,  symmetrically  reinforced  with  four  wires  4.4  mm.  (3^  in. 
approx.)  in  diameter,  were  subjected  to  tension,  and  the  stretch  both  in  the 
reinforcement  and  the  mortar  was  measured.  They  were  always  found  prac- 
tically equal.  From  the  known  modulus  of  elasticity  of  the  reinforcement,  and 
the  measured  stretch  of  the  steel,  could  be  computed  the  proportion  of  the  total 
tensile  stress  P,  carried  by  the  rein- 
forcement. The  remainder,  divided 
by  the  section  of  concrete,  gave  the 
unit  tension  in  the  mortar,  to  which 
its  measured  elongations  corres- 
ponded. 

The  observed  law  between  stress 
and  strain  is  shown  in  Fig.  50. 
The  ordinates  represent  the  total 
tensile  stress  on  the  prisms,  while  the 
abscissas  give  the  corresponding 
stretch  in  the  reinforcement.  As 
long  as  the  load  does  not  exceed 
a  certain  value,  Oa,  the  strains  in- 
crease uniformly  and  are  very  small. 

They  then  increase  suddenly,  but  soon  again  become  uniform  and  are  repre- 
sented by  the  flatter  straight  portion  AB  of  the  curve.  From  the  measured 
stretch  and  the  known  area  of  the  reinforcement,  the  part  of  the  load  carried 

*  Genie  Civil,  1899. 


52 


CONCRETE-STEEL  CONSTRUCTION 


by  it  can  be  calculated.  The  curve  for  the  steel  is  practically  a  straight  line  as 
long  as  the  elastic  limit  is  not  exceeded.  In  the  figure,  this  straight  line  is  repre- 
sented by  OF,  which  runs  practically  parallel  with  AB.  For  any  stretch  OP 
is  then 

PN  equal  to  the  part  of  the  load,  PM,  carried  by  the  steel, 
NM  "  "  concrete. 

It  therefore  follows  from  the  curve  that  the  concrete,  in  combination  with 
the  steel,  is  able  to  stretch  considerably,  but  that  after  a  certain  elongation  A, 
the  stress  on  the  concrete  does  not  materially  increase.  The  maximum  stretch 
was  0.9  mm.  (0.035  which  corresponds  with  a  steel  stress  of  1800  kg/cm^ 
(25,600  lbs/in^).  (This  is  less  than  the  first  value  of  2  mm.  per  meter,  found 
by  Considere).  The  lines  CB,  C'B',  C"B",  represent  repeated  loadings  and 
unloadings. 

Considere's  tests  were  repeated  by  the  French  Government  Commission  * 
with  somewhat  larger  prisms  of  1:2:4  concrete.  Similar  results  were  obtained, 
and  it  was  further  discovered  that  the  extensibility  of  reinforced  concrete  which 
had  set  under  water  was  greater  than  that  which  had  set  in  air. 

Considere's  tests  very  quickly  became  known,  and  were  at  once  used  by 
theorists  in  the  formation  of  new  methods  of  calculation,  without  waiting  for  con- 
firmatory experiments  or  even  considering  the  limitations  placed  by  Considere 
himself  on  the  practical  value  of  his  results. 

In  1904  objections  were  raised  to  Considere's  theory  by  both  American  and 
German  experimenters,  based  on  further  tests  made  by  them. 

The  experiments  of  A.  Kleinlogel,  conducted  in  the  Testing  Laboratory  of 
the  Royal  Technical  High  School  at  Stuttgart  were  published  in  Beton  und 
Eisen,"  No.  II,  1904,  and  also  No.  I  of  the  "  Forscherarbeiten  aus  dem  Gebiete 
des  Eisenbeton,"  Vienna,  1904.  They  comprised  rectangular  reinforced  con- 
crete beams,  220  cm.  (86.6  ins.)  long  and  15  by  30  cm.  (5.9  by  11.8  ins.)  in 
section.  The  mixture  was  i  cement:  i  sand:  2  crushed  limestone.  For  purposes 
of  comparison  some  beams  were  made  without  reinforcement.  The  beams  were 
supported  at  the  ends  and  loaded  with  two  symmetrically  placed  loads,  i  meter 
(39.4  ins.)  apart.  The  stretch  of  the  lowest  concrete  layer  was  measured  on  a 
length  of  80  cm.  (31.5  ins.),  included  within  the  central  portion  of  a  beam.  In 
order  to  make  the  cracks  more  evident,  the  lower  face  and  both  sides  of  the 
beams  were  painted  with  a  coat  of  whitewash.  The  six-months'  old  beams,  which 
had  been  kept  in  damp  sand,  gave  practically  equal  maximum  extensions  of 
the  lower  concrete  layer  for  several  different  percentages  of  reinforcement.  This 
amounted  to  between  0.148  and  0.196  mm.  per  meter  (0.000148  to  0.000196  ft. 
per  foot). 

Thus  Considere's  law  was  not  confirmed,  because  the  stretch  of  non-reinforced 
concrete  was  found  about  0.143  '^n^-  P^^  meter  (0.000143  ft.  per  foot).  (Accord- 
ing to  Considere,  it  was  o.i  to  0.2  mm.  per  meter.) 

Kleinlogel's  tests  also  furnished  important  information  about  adhesion,  tO' 
which  reference  will  be  made  later. 

*  Beton  und  Eisen,  No.  V,  1903. 


EXTENSIBILITY 


53 


Because  of  the  numerous  objections  raised  concerning  his  hypothesis,  Con- 
sidere  repeated  his  experiments  with  larger  specimens.*  The  concrete  consisted 
of  400  kg.  (880  lbs.)  of  Portland  cement,  0.4  cubic  meters  (0.52  cu.yds.)  of  sand, 
and  0.8  cubic  meters  (1.04  cu.yds.)  of  crushed  limestone.  The  beams  were 
of  rectangular  section,  3  meters  (9.84  ft.)  long,  15  cm.  (6  ins.)  wide,  and  20  cm. 
(7.8  ins.)  high,  and  were  reinforced  on  the  lower  side  with  two  round  rods  16 
mm.  (f  in.),  and  three  round  rods  12  mm.  (J  in.  approx.)  in  diameter.  As  in 
the  before-mentioned  experiments  they  were  tested  with  symmetrically  placed 
loads,  1.4  meters  (55  ins.)  apart,  within  which  distance  the  moment  was  uni- 
form and  no  lateral  forces  acted.  Of  two  specimens,  one  was  kept  under  damp 
sand,  and  one  under  water  for  six  months,  at  which  age  the  specimens  were 
tested.  It  was  shown  that  the  first  beam  stood  a  stretch 
between  the  layers  A  and  B,  of  from  0.22  to  0.5  mm.  (0.00866  ^ 
to  0.0196  in.),  and  the  second  (which  had  been  kept  under 
water),  stood  a  similar  stretch  of  from  0.56  to  1.07  mm. 
(0.022  to  0.04  in.).  Fig.  51.  A  crack  could  not  be  found  even  ^ 
though  the  outer  surface  was  coated  with  neat  cement.  The 
concrete  between  the  layers  A  and  B  was  sawed  out  and  still 
showed  the  same  strength  as  untouched  concrete.  Considere 
does  not  state  (and  such  is  the  case  with  all  his  tests)  whether  he  was 
able  to  cut  away  the  section  over  the  whole  length  of  the  beam  in  one  piece,  or 
whether  in  several. 

Of  the  experiments  conducted  by  the  Testing  Laboratory  of  the  Royal  Tech- 
nical High  School  at  Stuttgart,  concerning  the  extensibility  of  reinforced  con- 
crete, first  will  be  discussed  the 

Torsion  Tests  on  Hollow  Cylinders  with  Spiral  Reinforcement. — Hollow 
cylinders  of  the  same  dimensions  as  those  described  on  page  39  were  provided 
with  spiral  reinforcement  having  a  pitch  of  45°,  in  the  centers  of  the  walls. 
They  were  so  arranged  that  torsion  tests  would  produce  tension  in  the  spirals. 


Cylinder  IX,  with  five  spirals  of  7  mm.  (|  in.  approx.)  round  iron,  was 
tested  when  60  days  old.  Under  a  torque  if^  =  72,500  cm-kg  (62,800  in-lbs), 
two  cracks,  a  and  b  (Fig.  52),  at  right  angles  to  the  spirals,  were  observed.  The 
torque  was  increased  to  86,500  cm-kg  (74,900  in-lbs),  at  which  load  the  cracks 
opened  considerably. 

*  Beton  und  Eisen,  No.  Ill,  1905. 


54 


CONCRETE-STEEL  CONSTRUCTION 


In  Cylinder  X,  which  corresponded  exacdy  with  the  other,  a  fine  crack 
appeared  with  =  70,000  cm-kg  (60,600  in-lbs).  The  torque  was,  however, 
increased  to  120,000  cm-kg  (104,000  in-lbs)  when  further  parallel  cracks 
appeared. 

If  there  is  subtracted  from  the  value  for  Cylinder  IX  at  which  the  first  cracks 

appeared,  the  torque  =  54,560  cm-kg 
(47,200  in-lbs)  carried  by  the  non- 
reinforced  hollow  cylinders  of  equal  age, 
there  remains  in  Specimen  IX  the 
moment  1/"^  =  17,940  cm-kg  (15,600 
Fig.  53.  in-lbs). 

This  gives,  in  the  circle  of  21  cm. 
(8.27  in.)  diameter  in  which  the  spirals  lay,  a  total  horizontal  circumferential 
strength 

5  =  ^— =1710  kg  (3762  lbs.), 
10.  s  ^ 


half  of  which  must  be  taken  up  at  the  moment  of  cracking  by  the  reinforcement 
which  lies  at  an  angle  of  45°  to  this  theoretical  stress,  and  half  by  the  compres- 
sive resistance  of  the  concrete  acting  at  right  angles  to  the  direction  of  the  rein- 
forcement. Consequently,  from  Fig.  53,  Z=P=^a/ 2,  and  the  stress  in  the 
five  spirals  is 

.?55^         kg/cm2  (8960  lbs/in2). 

5X0.72^ 
4 


This  stress  may  also  be  obtained  from  the  torque  71/^  =  17,940,  by  a  proper 
distribution  of  the  inclined  tensile  stresses  r  over  the  section  of  the  reinforcement. 

For  cylinder  X,  the  steel  stress  at  the  appearance  of  the  first  crack  was  found 
to  be 

(7e  =  540  kg/ cm2  (7680  lbs/in^). 


With  Cylinder  XI,  with  10  spirals  of  10  mm.  (|  in.  approx.)  round  rods, 
otherwise  like  the  foregoing,  the  first  crack  {a)  appeared  at  =  125,000  cm-kg 
(108,200  in-lbs),  with  other  cracks  running  in  the  same  direction,  and  final 
rupture  at  1/^  =  155,000  cm-kg  (134,200  in-lbs). 

With  the  same  suppositions  as  before,  there  is  obtained  for  the  steel  stress 
at  the  appearance  of  the  first  crack 


in  Cylinder  XI,  ^^,  =  603  kg/cm2  (8580  lbs/in^); 


Xil,  (7c  =  56o     "      (7070     "  ). 


EXTENSIBILITY 


55 


It  is  thus  found  that  with  the  four  hollow  cylinders  the  first  cracks  in  the 
concrete  appeared  at  an  extension  which  corresponded  with  an  average  steel 
stress  of 

^^^630+540+603  +  560^^3^  kg/cm^  (8290  lbs/in2). 

4 

The  extension  at  this  stress  is  -=0.27  mm.  per  meter  (0.00027  ^t.  per  foot). 

2160 

If  the  shearing  stresses  at  the  appearance  of  the  first  crack  and  at  rupture 
are  computed  from  the  formula 

Md 
16  d 

the  results  of  Table  XXI  are  obtained: 


Table  XXI 

SHEARING  STRESSES  AT  FIRST  CRACK  AND  AT  RUPTURE 


Cylinder  No. 

At  the  First  Crack,  Td- 

At  Rupture. 

kg/cm^ 

lbs/in2 

kg/cm^ 

lbs/in2 

IX 
X 
XI 
XII 

25.2 
24.4 
43-6 
41.8 

358 
347 
620  • 

595 

30.2 
42.0 

49-5 
54-0 

430 
597 
704 
768 

It  may  be  concluded  from  this,  that  through  a  proper  arrangement  of  the 
reinforcement,  that  is,  by  placing  it  in  the  direction  of  the  maximum  tensile 
stresses,  the  shearing  strength  of  reinforced  concrete  can  be  increased  over  that 
of  plain  concrete. 

In  specimens  with  weak  reinforcement,  the  stress  at  rupture  rose  to  the  ulti- 
mate stress  in  the  steel;  while  with  heavier  reinforcement,  such  a  stress  could 
not  be  reached  because  the  adhesion  on  the  thicker  rods  was  not  sufficiently 
strong  at  the  ends. 

Bending  Tests  with  Reinforced  Beams  of  15  by  30  cm.  Section. 

These  specimens  had  the  same  dimensions  as  those  tested  by  Kleinlogel, 
but  were  made  with  i  cement  to  4  Rhine  sand  and  gravel.  They  were  constructed 
in  December,  1902,  and  tested  three  months  later  at  the  Testing  Laboratory  at 
Stuttgart.  They  were  consequently  older  than  Kleinlogel's  specimens.  They 
were  tested  with  two  symmetrically  placed  loads,  so  that  a  constant  moment 
(with  no  external  forces  acting),  was  obtained  throughout  the  central  portion 
of  80  cm.  (31.5  ins.)  between  the  loads.  Besides  the  stretch  of  the  steel,  the 
shortening  of  the  top  concrete  layer  was  also  measured,  and  the  deflection  within 
the  measured  length  was  also  ascertained  for  different  loads.  The  stretch  in 
the  steel  was  measured  between  projecting  lugs  A  A,  which  were  clamped  to 


56 


CONCRETE-STEEL  CONSTRUCTION 


the  reinforcement.  In  the  ends  of  the  beams,  the  two  reinforcing  rods  were 
arranged  as  shown  in  Fig.  54,  and  several  stirrups  were  provided  to  counteract 
the  local  effects  of  the  forces  P  and  the  shearing  and  adhesive  stresses.  These 
were  such  that  no  cracks  appeared  between  the  supports  and  the  loads  P. 

The  six  specimens  were  severally  reinforced  with  two  10  mm.  (|  in.  approx.), 
with  two  16  mm.  (|  in.),  and  with  two  22  mm.  (|  in.)  rods.  Of  these  beams, 
three  were  used  for  the  determination  of  the  steel  stretch,  and  three  for  the 
shortening  of  the  top  concrete  layer,  because  the  apparatus  w^as  so  designed  that 
both  observations  could  not  be  made  simultaneously. 

The  tension  face  of  each  beam  received  a  coat  of  whitewash  to  make  the 
cracks  easier  of  discovery.  The  first  cracks  z  v/ere  always  noted  next  the  lugs 
A,  probably  because  at  those  points  the  zone  of  tension  in  the  concrete  was  weak- 
ened. Afterward,  the  cracks  m,  n^,  and  no,  appeared  within  the  central  portion. 
All,  indeed,  were  so  minute  that  they  probably  would  not  have  been  seen  except 
for  the  coat  of  whitewash. 

From  the  stretch  in  the  plane  of  the  reinforcement,  and  the  shortening  of 
the  top  layer,  the  extensibility  of  the  lowest  layer  could  be  computed.    The  tests 


gave  the  values  shown  in  Table  XXII,  at  which  the  cracks  appeared  within 
the  measured  length: 

Table  XXII 
EXTENSIBILITY  EXPERIMENTS 


Reinforcement. 

Stretch  of  the  Steel. 

Stretch  of  Lowest  Con- 
crete Layer. 

Number  of 
Round 
Rods 

Diameter. 

Per  Cent. 

.  mm. 

in. 

mm/m 

ft/foot 

mm/m 

ft/foot 

2 

10 

1 

0.4 

0.42 

0.00042 

0.50 

0.00050 

2 

16 

5 

8 

1 .0 

0-33 

0.00033 

0.40 

0.00040 

2 

22 

I 

1.9 

0.30 

0.00030 

0.38 

0.00038 

This  was  about  treble  that  of  non-reinforced  concrete.  After  the  specimens 
were  prepared,  they  w^re  kept  moist  for  a  considerable  time,  but  were  tested 
in  an  air-dry  condition.  The  difference  between  Considere's  tests  and  those 
of  other  experimenters  can  be  partially  explained,  since  concrete  which  sets 
under  water  swells  and  therefore  stands  greater  stretching  than  that  which  sets 
in  air  and  decreases  in  volume.  It  is  also  to  be  noted  that  with  each  repetition 
of  his  experiments,  Considere  found  smaller  results.    From  2  mm.  they  fell  to 


EXTENSIBILITY 


57 


o.Q  mm.,  and  finally  to  0.5  mm.  per  meter  (from  0.0020  to  0.0005  P^^  foot). 
The  latter  figure  does  not  differ  much  from  the  results  on  pages  53  to  55. 

These  bending  tests  will  be  discussed  again  later,  in  connection  with  the 
subject  of  the  exact  location  of  the  neutral  axis  and  the  distribution  of  stress 
in  the  section.  Also,  there  will  be  given  an  independent  explanation  of  the 
large  extensibility  observed  by  Considere  and  of  the  stress  distribution  between 
steel  and  concrete,  shown  in  Fig.  50.  A  com|)lete  statement  is  imi)ossible  with- 
out having  first  discussed  the  theory  of  reinforced  concrete. 

Similar  experiments  were  carried  out  for  the  Reinforced  Concrete  Com- 
mission of  the  Jubiliiumsstiftung  der  Deutschen  Industrie  in  the  Testing  Labora- 
tory at  Stuttgart.  In  them,  Bach  *  thoroughly  investigated  the  appearance 
of  the  first  crack  in  beams  of  which  the  material,  proportions,  and  load  distri- 
bution were  similar  to  those  illustrated  in  Fig.  54,  and  the  outside  of  which  was 
given  a  coat  of  whitewash.  With  increasing  load  on  the  under  side  of  the  beams, 
small  damp  spots  first  showed  themselves.  These  spots  grew  in  size  as  the  load 
was  augmented.  With  further  increase,  cracks  appeared,  always  where  a  spot 
of  water  existed,  but  not  all  such  spots  developed  into  cracks.  These  phenomena^ 
which  had  been  described  by  Turneaure,  "  Engineering  News,"  1904,  p.  213, 
and  also  by  R.  Feret,  "Etude  experimentale  du  ciment  arme,"  1906,  developed  in 
beams  which  had  been  kept  under  water,  and  may  be  explained  by  their  porosity 
in  certain  portions  which  were  stretched  by  the  tensile  stresses  and  from  which 
the  moisture  w^orked  outward  and  so  formed  the  spots  of  water  on  the  surface. 
The  cracks  appeared  on  the  sides  of  the  beams  at  somewhat  higher  loads  than 
on  the  bottom.  It  was  further  shown  that  the 
cracks  usually  commenced  at  the  bottom  corner, 
furthest  from  the  reinforcement.  In  the  section 
shown  in  Fig.  55  a  crack  existed  at  a  load  of  6000 
to  6500  kg.  (13,200  to  14,300  lbs.)  at  depths  about 
as  shown  by  the  lines  ah  and  cd,  and  advanced  under 
a  load  of  7000  kg.  (15,400  lbs.)  to  the  positions  of 
ai^i,  Cidi.  In  the  beams  with  a  single  reinforcing 
rod  the  cracks  appeared  somewhat  later  in  the 
narrower  beams  than  in  the  wider  ones.  The  first 
corner  crack  was  observed  at  a  stretch  of  from  0.127- 

0.176  mm.  in  a  length  of  one  meter  (0.0001 27-0.0001 76  ft.  per  foot)  for  a  beam 
15  to  30  cm.  (5.9  to  11.8  ins.)  wide  with  a  single  reinforcing  rod.  The  spots  of 
moisture  always  appeared  with  a  stretch  of  0.08-0.10  mm.  per  meter  (0.00008 
to  0.00010  ft.  per  foot),  depending  on  the  distribution  of  the  steel  in  the  section. 
This  is,  however,  the  ultimate  .stretch  of  plain  concrete.  The  formation  of 
cracks  will  be  delayed  if  the  reinforcement  in  the  vicinity  of  the  porous  spots 
in  the  stretched  concrete  receives  additional  assistance.  When  the  reinforce- 
ment was  uniformly  distributed  over  the  whole  width  of  the  beam,  the  cracks 
were  actually  found  after  greater  stretching,  but  were  much  smaller  and  cor- 
respondingly harder  to  discover.    In  heavily  reinforced  beams  the  extension 


*  "  Versuche  mit  Eisenbelon-Balken,"  Part  I.,  No.  39  of  the  Mitteilungen  iiber  Forschungs 
arbeiten  and  No.  26  of  the  Zeitschrift  des  Vereins  Deutscher  Ingenieure,  1907. 


58 


CONCRETE-STEEL  CONSTRUCTION 


at  which  the  first  crack  appeared  was  correspondingly  increased  to  0.267  "Sni- 
per meter  (0.000267  ft.  per  foot).  The  maximum  stretch,  which  amounted 
to  0.324  mm.  (0.000324  ft.)  for  beams  stored  in  moist  sand,  and  0.367  mm. 
(0.000367  ft.)  for  those  in  water,  was  found  in  beams  in  which  the  reinforce- 
ment was  in  the  form  of  a  plate  7  mm.  (J  in.  approx.)  thick  containing  holes 
and  extending  across  the  full  width  of  the  beam. 

The  influence  on  the  beams  of  dry  and  wet  storage  was  also  investigated. 
Beams  of  30  by  30  cm.  (12  by  12  in.  approx.)  section,  provided  on  the  under 
:side  with  one  round  rod  26  mm.  (i  in.  approx.)  in  diameter,  stored  in  air, 
stretched  0.097  mm.  (0.000097  ft.),  but  those  stored  under  water  stretched  0.205 
mm.  per  meter  (0.000205  ft.  per  foot)  before  the  appearance  of  the  first  crack. 
Since  non-reinforced  concrete  which  has  been  stored  under  water  or  in  a  moist 
condition  swells,  reinforced  concrete  which  sets  under  water  must  develop 
tensile  stress  in  the  steel,  and  corresponding  compressive  and  bending  stresses  in 
the  concrete.  These  compressive  stresses  are  naturally  greatest  in  the  layers  near 
the  reinforcement,  and  it  is  clear  that  when  a  load  is  applied  it  first  overcomes 
these  compressive  stresses  in  the  concrete  and  hence  the  stretch  up  to  the 
first  crack  is  greater  than  when  concrete  originally  in  an  unstressed  state,  is 
stretched.  In  dry  concrete  which  has  set  in  the  air,  a  reduction  in  volume  or 
shrinkage  takes  place,  so  that  in  this  case  in  unstressed  beams  the  steel  is  com- 
pressed and  the  concrete  subjected  to  tension  and  bending.  Here  the  first 
crack  will  appear  under  less  load  and  shorter  stretch. 

The  experiments  with  T-beams,  which  will  be  described  later,  also  contribute 
something  concerning  the  extensibility  of  concrete. 

Methods  of  calculation  will  next  be  considered,  and  in  connection  with  them 
will  be  given  further  experiments  on  reinforced  concrete  bodies,  so  that  the 
methods  of  computation  can  be  checked  by  them. 


CHAPTER  V 
THEORY  OF  REINFORCED  CONCRETE 
COMPRESSION 

Calculation  of  Reinforced  Concrete  Columns  with  Longitudinal  Rods  and 

Ties. — In  homogeneous  bodies,  subject  to  axial  compressive  stress,  it  is  assumed 
that  the  resulting  strain  takes  place  by  a  lessening  of  the  distance  between 
adjacent  imaginary  parallel  planes  perpendicular  to  the  line  of  stress,  and  in 
such  manner  that  the  planes  remain  mutually  parallel  after  the  strain.  This 
same  assumption  is  also  made  in  the  calculation  of  strains  in  concrete  columns 
with  longitudinal  reinforcement  if. 

First,  that  portion  of  the  axial  stress  borne  by  the  concrete  is  assumed  as 
uniformly  distributed  over  the  whole  cross-section,  and 

Second,  the  reinforcement  has  the  same  deformation  as  the  concrete. 

If  represents  the  area  of  the  concrete  cross-section,  the  total  area  of  the 
reinforcement,  cr^  and  the  corresponding  stresses  in  the  two  materials  when 
equally  strained,  the  total  load  P  will  be 

In  the  design  of  any  new  column,  either  the  experimentally  determined  stress- 
strain  curve  or  the  exponential  law,  £^=a'(7j,^,  may  be  employed  in  selecting 
corresponding  values  of  and  a^,  which  can  then  be  inserted  in  the  general 
load  formula.  On  the  other  hand  it  is  only  by  the  method  of  approximations, 
or  of  interpolation  in  tables,  that  it  is  possible  to  find  the  exact  stresses  in  an 
existing  column.    In  the  "Leitsatze"  of  the  Verbands  Deutscher  Architekten-  und 

E 

Ingenieurvereine,  the  ratio  -^=^  =  15,  is  assumed  as  constant,  so  that  with 

Fh 

equal  strains  on  steel  and  concrete, 

Ee 

and  the  column  load  is 


P=--oh{Fb  +  isFe). 


59 


60 


CONCRETE-STEEL  CONSTRUCTION 


As  the  safe  stress  on  concrete  is  assumed  at  cr;,  =  35  kg/cm^  (497  lbs/in^), 
it  follows  that  the  safe  load  on  a  reinforced  concrete  column  is 

from  which  may  be  derived 

In  given  columns,  the  unit  stresses  will  be 

Ee  . 

The  ratio  is  less  than  15  within  the  limits  of  perfect  elasticity,  being 

Eb 

approximately  10,  but  the  higher  value  was  chosen  in  the  "  Leitsatze  "  so  as  to 
take  account  of  conditions  near  rupture. 

One  point  with  regard  to  reinforced  concrete  design  deserves  the  greatest 
consideration.  With  homogeneous  materials,  the  dimensions  of  pieces  are 
usually  determined  from  safe  stresses  which  are  definite  fractions  of  the  ultimate 
loads.*  In  reinforced  concrete,  however,  the  question  arises  whether  the  allow- 
able and  assumed  load  distribution  existing  with  safe  stresses  still  continues  near 
the  point  of  rupture,  or  whether  conditions  change  so  that  the  real  causes  of  rup- 
ture are  different,  just  as  is  the  case  in  computations  with  regard  to  allowable 
tension  in  long  columns.  Nothing  but  experiments  can  afford  information  about 
these  important  questions. 

Until  1905  compression  tests  of  columns  were  very  rare,  although  great 
responsibility  is  involved  in  their  design  and  construction. 

A  column  with  4^  per  cent  of  reinforcement  was  tested  at  the  Technical 
High  School  in  Charlottenburg.  Its  sectional  dimensions  were  25  by  25  cm. 
(10  by  10  in.  approx.),  and  height  3.22  m.  (127  in.).  The  reinforcement  was 
4  round  rods,  30  mm'.  (13^  in.)  in  diameter,  which  were  connected  at  50  cm. 
(20  in.)  intervals  by  horizontal  flat  iron  ties  3  mm.  by  80  mm.  (J  by  3  in. 
approx.) ;  the  mixture  was  i :  4,  age  3  months.  The  column  \vas  prepared  with 
accurate  compression  surfaces,  and  failed  in  such  manner  that  the  four  rods 
buckled  simultaneously  between  two  ties,  and  the  concrete  between  them  crushed. 
The  breaking  strength  was  255  kg/cm^  (3627  lbs/in^). 

If  the  reinforcing  of  concrete  columns  with  longitudinal  steel  and  horizontal 
ties  is  so  done  as  to  secure  at  least  as  much  strength  in  the  long  members  as  in 
test  cubes,  then  it  is  necessary,  in  designing,  only  to  consider  the  strength  of 
short  specimens  (or  a  certain  part  of  such  strength),  and  the  load  is  P  =  FbOb. 
The  steel  would  then  be  entirely  omitted  from  consideration,  but  at  the  same 

*  Exceptions,  however,  exist.  For  instance,  the  computation  of  the  flexure  at  the  breaking 
load  will  fail.  The  method  of  calculation  of  beams  of  the  Schwedler  type  of  construction  is 
inaccurate,  wherein  the  load  is  assumed  greater  so  as  to  include  the  necessary  safety  in  regard 
to  diagonal  tension.    Also  the  computation  of  retaining  walls  and  chimneys  as  to  overturning. 


COMPRESSION 


61 


time  enough  must  be  employed  in  the  form  of  longitudinal  reinforcement  and 
ties,  so  that  the  breaking  load  of  a  reinforced  concrete  column  will  be  equal  to 
that  of  a  small  cube.  It  is  evident  that  a  certain  minimum  of  steel  is  necessary. 
In  the  "  Leitsiitze,"  longitudinal  reinforcement  not  less  than  0.8  per  cent  of  the 
cross-section  is  prescribed. 

But  the  spacing  of  the  ties  also  influences  the  breaking  load  of  a  column. 
Their  effect  is  even  greater  than  that  of  the  longitudinal  rods,  as  was  proved 
by  the  latest  experiments  of  the  Reinforced  Concrete  Commission  of  the  Jubil- 
aumsstiftung  der  Deutschen  Industrie. 

These  tests,*  made  in  1905,  were  conducted  by  Bach  at  the  Testing  Lab- 
oratory of  the  Royal  Technical  High  School  of  Stuttgart,  and  involved  concrete 
prisms  25  by  25  cm.  (10  by  10  in.  approx.)  in  section,  and  i  meter  (39.4  in.) 
long,  mixed  in  the  proportions  of  i  part  Portland  cement  and  4  parts  of 
Rhine  sand  and  gravel,  with  15  per  cent  of  water.  Thus  they  were  of  the  same 
composition  as  the  specimens  described  on  pages  44  to  46,  used  in  the  adhesion 
experiments. 

Part  of  the  specimens  were  without  reinforcement.  The  others  each  had 
4  rods  with  7  mm.  (J  in.  approx.)  ties  arranged  as  shown  in  Fig.  56.  Five  varie- 
ties of  reinforcement  were  employed,  viz.: 

15  mm.  (f  in.)  rods  and  25  cm.  (10  in.)  tie  spacing,  Fig.  57 
15  mm.      in.)  rods  and  12.5  cm.  (5  in.)  tie  spacing,  Fig.  58 
15  mm.  (I  in.)  rods  and  6.25  cm.  (2^  in.)  tie  spacing,  Fig,  58 
20  mm.  (I  in.)  rods  and  25  cm.  (10  in.)  tie  spacing,  Fig.  58 
30  mm.  (i^V  in.)  rods  and  25  cm.  (10  in.)  tie  spacing,  Fig.  59 

At  the  same  time  was  ascertained  the  compressive  strength  of  cubes,  30  cm. 
(12  in.)  on  each  edge. 

The  elasticity  in  compression  was  measured  for  two  or  three  specimens  of 
each  kind,  for  stresses  up  to  113  kg/cm^  (1607  lbs/in^).  It  was  disclosed  that 
the  shortening  diminished  not  only  with  increased  section  of  longitudinal  steel, 


^  -  

"  r-  ™= 



1.  ..>r..iir.riti>>k,.  ^^,L,..la 

Fig.  56. 

but  also  with  increasing  numbers  of  ties,  when  the  longitudinal  rods  were  the 
same.  In  the  same  manner  as  described  on  page  21,  for  stress  increments  of 
about  16  kg/cm2  (228  lbs/in^)  were  measured,  the  total  compression,  the  elastic 
deformation,  and  the  permanent  set.  From  these  results,  curves  were  determined 
similar  to  those  for  plain  concrete.  The  influence  of  the  ties  on  the  elastic  phe- 
nomena is  shown  in  Table  XXIII. 

*  C.  V.  Bach  "  Druckversuche  mit  Eisenbetonkorpera,"  1905.  "  Mitteilungen  iiber  For- 
schungsarbeiten,"  No.  29. 


62 


CON(^RETE-STEEL  CONSTRUCTION 


Fig.  57.  Fig.  58.  Fig.  59. 

Test  Specimens. 


Table  XXIII 


ELASTICITY  TEST   OF  COLUMNS 


Stresses 

Diameter  of  Rods 

Tie  Spacing 

Shortening  in  Millionths  of  the 
Length 

kg/cm2 

lbs/in2 

mm. 

in. 

cm. 

in. 

Total 

Elastic  Dif. 

Permanent 
Set 

32.3 
32.3 
32-3 
32-3 

459 
459 
459 
459 

Plain 

concrete 

133 
114 
no 
106 

7 

5 
2 

4 

126 

15 
15 
15 

f 
5 

8 
5 
8 

25 

12.5 
6.25 

ID 

5 

2i 

109 
108 
102 

64.6 
64.6 
64.6 
64.6 

919 
919 
919 
919 

Plain 

concrete 

333 
267 
264 
241 

37 
20 
18 
13 

296 

247 
246 
228 

15 
15 
15 

1 
5 

8 
5 
8 

25 

12.5 
6.25 

ID 

5 

2i 

97.0 
97.0 
97.0 
97.0 

1380 
1380 
1380 
1380 

Plain 

concrete 

709 
488 

473 
421 

164 

63 
58 
42 

545 
425 
415 
379 

15 
15 
15 

1 
5 

8 
f 

25 

12.5 
6.25 

ID 

5 

2h 

COMPRESSION 


63 


It  is  there  shown  that,  even  with  known  elastic  data  for  plain  concrete,  it  is 
impossible  to  determine  the  distribution  of  stress  between  the  steel  ana  concrete 
with  usual  stresses,  since  the  ties  alter  the  elasticity  of  the  reinforced  concrete. 
They  prevent  lateral  expansion  of  the  concrete  and  thereby  increase  its  com- 
pressive strength.  The  assumed  ratio  of  the  moduli  of  elasticity  of  steel  and 
concrete  is  a  close  enough  approximation.  Actually  it  varies  from  i:ii  to  1:13 
at  the  highest  stresses  covered  by  the  elasticity  experiments. 

For  practical  purposes,  the  observed  breaking  strengths  are  more  important. 
(See  Table  XXIV.) 


Table  XXIV 
BREAKING  STRENGTH  OF  COLUMNS 


Specimen  about  3  Months  Old 


Diameter  of  Rods 


Tie  Spacing 


Breaking  Strength 


Each 


Average 


kg^cm2       lb /in  2 


%of 
Rein- 
forcement 


15 
15 
15 
20 


Plain 


Test 


concrete 
25 

12.5 
6.25 

25 
25 

cubes 


10 
5 

2^ 
10 
10 


146 
171 
168 
212 
169 
174 

I  168 


138 

139 

161 

172 

187 

175 

200 

203 

169 

172 

199 

197 

169 

171  \ 

185 

184  / 

141 
168 
177 
205 
170 

190 

175 


2010 

2390 
2520 

2920 
2420 
2700 

2490 


o 

1. 14 
1. 14 
1. 14 

2.04 
4.60 


The  appearance  at  fracture  is  shown  in  Figs.  60  to  64.  According  to  the 
*'Leitsatze"  of  the  Verbande  Deutscher  Architekten-  und  Ingenieur-Vereine,  the 
allowable  loads  for  the  prisms  were  as  in  Table  XXV. 


Table  XXV 

ALLOWABLE  COLUMN  LOADS 

With  four  15  mm.  (|  in.)  rods,  P  =  625X35+i5X  7.1X35=25602  kg.  (56324  lbs.) 
With  four  20  mm.  (fin.)  rods,  P  =  625X  35+ 15X  12.6X35=  28490  kg.  (62678  lbs.) 
With  four  30  mm.  (i^  in.)  rods,  ^=625X35+ 15X  28.3X35  =  36732  kg.  (83010  lbs.) 

These  computed  allowable  loads  are  in  the  proportion  of  168:187:241,  while 
the  actual  loads  on  specimens  with  a  25  cm.  (10  in.)  tie  spacing  were  as 

168: 170: 190. 

It  is  seen  from  these  values  that  an  increase  in  the  area  of  longitudinal  rein- 
orcement  does  not  produce  an  increase  in  the  breaking  strength  to  the  extent 
which  would  be  indicated  by  the  formula 


64 


CONCRETE-STEEL  CONSTRUCTION 


In  experienced  hands  this  formula  may  give  rise  to  constructions  which  are 
not  sufficiently  safe.  Some  designers  are  careless  with  regard  to  this  point, 
and,  in  order  to  produce  columns  of  small  diameter,  increase  the  percentage 
of  longitudinal  reinforcement  disproportionately.  This  gives  such  columns  a 
calculated  margin  of  safety  which  they  do  not  possess. 

When  the  increase  in  resistance  is  computed  for  one  kilogram  of  steel  in  the 
form  of  longitudinal  rods  and  of  ties,  it  is  discovered  that  the  steel  used  as  ties 
is  nearly  twice  as  effective  as  the  straight  rods.  The  former  must,  therefore, 
be  given  proper  attention  m  the  design  of  columns.  Further  experiments  with 
long  columns,  in  which  the  top  and  bottom  are  broadened  to  guard  against 
premature  failure,  would  be  very  desirable,  and  are  planned  by  the  Reinforced 
Concrete  Commission. 


Fig.  6o.  Fig.  6i,  Fig.  62.  Fig.  63.  Fig.  64. 

Results  of  Crushing  Tests. 


It  is  recommended,  until  further  tests  are  available,  in  reinforced  concrete 
building  construction  designed  in  accordance  with  the  formulas  of  the  German 
''Leitsatze,"  that  20  kg/cm^  (284  lbs/in^)  be  assumed  for  the  columns  of  the  upper 
floor,  and  that  this  stress  be  increased  in  the  lower  floors  to  the  maximum  safe 
limit.  The  longitudinal  reinforcement  should  be  from  0.8  to  2.0  per  cent,  and 
the  tie  spacing  approximately  5  cm.  (2  in.)  less  than  the  diameter  of  the  column, 
but  never  over  35  cm.  (13.8  in.).  If  the  strength  of  test  cubes  is  made  the  basis 
for  assumed  safe  loading  without  at  all  considering  the  steel,  then  the  stresses 
may  range  from  25  kg/cm^  (355  lbs/in-)  in  the  top  story  to  45  or  50  kg/cm^ 
(64c  to  710  lbs/in^)  in  the  lower  ones.  It  is  hardly  imaginable  that  all  floors 
of  a  building  will  ever  be  simultaneously  fully  loaded,  so  that  the  columns  of 
the  lowest  story  are  very  seldom  fully  stressed,  and  consequently  are  most  favored. 

The  computation  of  the  tie  spacing  solely  from  the  buckling  length  of  the 
longitudinal  steel,  too  often  calls  for  excessive  spacings.    Furthermore,  in  prac- 


COMPRESSION 


65 


tice,  the  actual  spacings  are  not  mathematically  exact,  and  do  not  remain  fixed 
during  the  tamping  of  the  concrete.  They  should  prevent  lateral  Imlging  of 
the  concrete,  and  must  therefore  possess  ample  strength  to  resist  lateral  failure. 

Flexure. — No  tests  of  the  breaking  strength  of  reinforced  concrete  columns 
exist,  com])arable  with  those  for  steel.  It  therefore  becomes  necessary,  in  design- 
ing in  reinforced  concrete,  to  em])loy  data  applicable  to  all  homogeneous  bodies, 
Tetmajor  shows  that  Euler's  formula 


applies  to  long,  slender  steel  columns,  if  the  compressive  stress  lies  below  the 
elastic  limit  w^hen  failure  commences.  To  large  cross-sections  and  short  lengths, 
this  formula  probably  does  not  apply,  because  the  compressive  stress  at  the 
breaking  point  has  exceeded  the  elastic  limit.  Under  such  circumstances  E,  the 
modulus  of  elasticity,  is  not  a  constant.    In  all  materials  such  as  concrete  which 


ZOO 


Fig.  65. 


Fig.  66. 


•do  not  possess  a  constant  modulus,  it  is  necessary  to  ascertain  (by  calculation 
from  experiments),  a  value  for  E  which  will  correspond  with  the  compressive 
stress  at  the  moment  of  rupture,  in  such  manner  that  it  may  be  measured  by  the 
tangent  of  the  angle  of  inclination  of  the  stress-strain  curve  and  be  expressed  by 


da 


In  calculating  the  area  of  the  cross-section  involved  in  the  moment  of  inertia  /, 
the  area  of  the  steel  must  be  multiplied  by  the  ratio  EJEf^.  No  change  in  the 
distribution  of  stress  can  take  place  with  such  procedure,  since  the  area  of  the 
reinforcement  is  replaced  by  a  concrete  area  E^IE^  times  as  large. 

Since  the  exponential  law  of  stress-strain  variation  applies  only  to  stresses 
up  to  about  40  kg/cm^  (568  lbs/in^),  neither  can  that  law  be  utiHzed  for  the 
derivation  of  a  suitable  breaking  formula. 


66 


CONCRETE-STEEL  CONSTRUCTION 


In  the  1899  volume  of  the  Schweizerische  Bauzeitung,  is  a  communication 
by  Ritter  giving  another  basis  for  a  formula.  In  it  the  following  equation  is 
employed: 

(7  =  X(l-e-1000.S)^ 

K  represents  the  ultimate  stress  in  the  concrete,  E  the  corresponding  strain,  and 
€  =  2.71828,  the  base  of  the  system  of  natural  logarithms.    If  the  locus  of  this 
equation  is  plotted,  the  curve  will  be  found  to  agree  as  well  as  can  be  expected 
with  the  stress-strain  curves  found  by  experiment  under  varying  circumstances. 
By  differentiating  g  with  respect  to  e,  the  modulus  of  elasticity  E  is  obtained. 

£=^  =  XlOOOC-1000^  =IOOO(A'-(7). 

If  this  value  is  inserted  in  Euler's  formula,  there  results 


P=-j^EJ=—iooo{K—a)J; 


a  therefore  represents  the  initial  breaking  stress.  If  P  is  replaced  by  Fa,  J  by  Fr^ , 
and  71^  by  10,  there  is  obtained  for  the  breaking  stress 

K 


I  +0.0001—^ 


Example.  Let  it  be  desired  to  obtain  the  breaking  load  on  a  column  having 
a  cross-section  of  25  by  25  cm.  (9.84  by  9.84  in.)  reinforced  with  4  rods  18  mm. 
(0.708  in.)  in  diameter, 

Ee 

/  =  4m.  (157.48  in.),    ;g^  =  io.    ^  =  250  (3555), 
7=^25^  + 10X4X0.9^X7: X  10^=42702  cm^, 

(=^9-8^  +  ioX4Xo.354'X7rX3W  =  i025  in^); 

F  =  252  +  ioX4Xo.92x-/r  =  727  cm^, 
(  =  9.82  +  10X4X0.354 X27:  =  iT3  in2); 

r2  =  -^  =  58.7  cm2  (9.1  in^); 
t 


COMPRESSION 


67 


Ok 


250 


I  +0.0001  X 


400 

78T 


->  =  i97^kg/cm2, 


3555 


I  +0.0001  X 


I57-48- 
9.1 


=  2802  lbs/in2). 


With  a  factor  of  safety  against  rupture  of  eight,  a  safe  working  compressive 
stress  of  25  kg/cm^  (355  lbs/in^)  can  be  assumed.  If  partially  fixed  ends  are 
considered,  a  corresponding  free  length  of  j  /  should 


be  employed,  and 


closer  result 


will  be  somewhat  larger.  A 
be    obtained  with 


might 


somewhat 

W  =  II. 

In  the  example  above,  a  comparatively  slender 
column  has  been  assumed.  It  is  evident  from  the 
result  of  the  calculation  that  reinforced  concrete 
columns  differ  very  materially  from  iron  ones,  the 
risk  of  breakage  being  much  greater  in  the  latter. 
The  superiority  of  reinforced  concrete  is  due  to 
the  greater  sectional  area  (as  compared  with  steel) 
and  the  smaller  unit  stress.  Or,  in  terms  of  the 
Euler  formula,  the  moment  of  inertia  /  is  increased 
in   greater    ratio   than  the  modulus  of  elasticity  is 

reduced,  when  compared  with  a  steel  column  of  equal  carrying  capacity. 

Consequently,  with  concrete,  only  in  exceptional  cases  will  there  be  required 
a  special  calculation  of  the  safety  against  rupture  by  flexure. 


Fig.  67. 


Calculation  of  Reinforced  Concrete  Columns  with  Spiral  Reinforcement 

(Beton  Frette)* 

Considere's  method  of  calculating  spirally  reinforced  columns  will  here  be 
followed.  From  theoretical  considerations,  the  correctness  of  which  has  been 
fully  established  by  experiment,  Considere  reached  the  conclusion  that  rein- 
forcement, if  introduced  in  the  form  of  a  spiral,  ensured  an  increase  in  the 
carrying  capacity  2.4  times  as  great  as  would  be  obtained  with  the  same  amount 
of  reinforcement  in  the  shape  of  longitudinal  rods. 

If  Fb  represents  the  area  of  the  concrete  core,  k  the  ultimate  stress  of  non- 
reinforced  concrete,  the  cross-section  of  the  longitudinal  reinforcement,  J  J  the 
cross  section,  of  imaginary  longitudinal  rods,  of  which  the  weight  is  equal  to 
that  of  the  spirals  in  an  equal  length  of  column,  Oe  the  elastic  limit  of  the  rein- 

*  Patented  in  France,  Germany,  England,  United  States,  etc. 


68 


CONCRETE-STEEL  CONSTRUCTION 


forcement  (which,  for  commercial  material,  may  be  assumed  at  2400  kg/cm^ 
(34,140  lbs/in^)),  then  the  ultimate  load  is  given  by  the  formula 

I.5^F6  +  ^7,(/,  +  2.4//). 

In  this  expression,  it  is  supposed  that  the  elastic  limit  of  the  reinforcement 
determines  the  carrying  capacity  of  the  column.  The  factor  1.5  is  employed 
because  an  octagonal  cross-section,  together  with  other  usual  conditions,  make 
the  gross  sectional  area  about  equal  to  1.5  times  the  central  portion  enclosed  by 
the  'spiral.    It  thus  equals  1.5^5  of  the  whole  concrete  section. 

Considere  *  proved  that  test  specimens,  prepared  with  the  care  possible  in 
a  laboratory,  developed  very  considerable  compressive  strength.    The  owners 


Fig.  68.  Fig.  69.  tiG.  70.  Fig.  71. 


Test  Specimens. 

of  the  German  rights  under  Considere's  patents,  deemed  it  advisable  to  institute 
experiments  with  specimens  manufactured  without  special  care,  at  a  building 
site.  As  a  consequence,  in  the  specimens  so  prepared,  the  pitch  of  the  spirals 
was  rendered  somewhat  irregular  by  the  ramming  of  the  concrete,  and  some 
eccentricity  of  position  was  perceptible.  In  the  earlier  tests  the  longitudinal 
reinforcement  had  a  sectional  area  of  at  least  one  per  cent  of  that  of  the  specimen, 
with  a  pitch  in  the  spirals  of  one-seventh  of  the  diameter  of  the  column.  The 

*  Genie  Civil.,  Nov.,  1902,  Beton  und  Eisen,  No.  V,  1902. 


COMPRESSION 


69 


later  experiments  were  intended  to  show  whether  it  is  advisable  to  increase  the 
pitch  of  the  spirals  appreciably  beyond  this  ratio. 

The  specimens  had  an  octagonal  section  with  a  diameter  of  27.5  cm.  (10.8  in.) 
and  a  length  of  i.oo  m.  (39.37  in.),  and  were  made  of  a  mixture  of  i  part  Ijy 
volume  of  Heidelberg  Portland  cement,  to  4  parts  of  Rhine  sand  and  gravel, 
with  14  per  cent  (by  volume)  of  water.  They  were  between  5  and  6  months  old 
when  tested.  Three  specimens  of  each  kind  were  broken.  Their  general  dimen- 
sions are  given  in  Figs.  68  to  71  and  Table  XXVI. 

Table  XXVI 


DIMENSIONS   OF  CONSIDER?:  COLUMN  TESTS 


Spiral 

Longitudinal  Reinforcement 

ISIO. 

Average 
of  of 

Figure 

Thickness  d 

Pitch  s 

Diameter 

Specimen 

No.  of 

Rods 

mm. 

in. 

mm. 

in. 

m.m 

in. 

I 

4 

68 

None 

None 

None 

None 

II 

3 

69 

5 

3 

ic 

38 

1.50 

4 

7 

1 

III 

3 

69 

7 

1 
4 

37 

1 .46 

4 

7 

1 
4 

IV 

3 

69 

10 

f 

42 

1.62 

4 

7 

1 
4 

V 

3 

70 

5 

3 

10 

38 

1-50 

8 

II 

7  • 

16 

VI 

3 

70 

7 

i 

37 

1 .46 

8 

1 1 

7 

10 

VII 

3 

70 

10 

3 

8 

43 

1 . 70 

8 

II 

1^ 

VIII 

3 

69 

7 

1 

31 

1 . 22 

4 

7 

1 
4 

IX 

3 

69 

10 

3 
8 

40 

1.58 

4 

7 

1 

X 

3 

69 

12 

41 

1.62 

4 

7 

XI 

3 

69 

14 

37 

1.46 

4 

7 

i 

XII' 

3 

70 

7 

1 
i 

40 

1.58 

8 

5 

XII" 

3 

70 

10 

*9 

40 

T.58 

8 

7 

1 

4 

XII'" 

3 

70 

14 

16 

40 

1.58 

8 

10 

f 

XIII' 

3 

71 

7 

1 
4 

80 

3-16 

8 

7 

i 

XIII" 

3 

71 

10 

f 

80 

3-16 

8 

10 

t 

XIII'" 

3 

71 

14 

9 

TTi 

80 

3.16 

8 

12 

XIV' 

3 

71 

7 

1 

120 

4.72 

8 

10 

1 

XIV" 

3 

71 

10 

f 

1 20 

4-72 

8 

12 

XIV'" 

3 

71 

14 

120 

4.72 

8 

14 

9 

In  some  of  the  tests  *  the  permanent  and  elastic  deformations  were  both 
measured,  but  no  definite  law  could  be  deduced  from  the  results  except  that 
the  reinforced  columns  displayed  somewhat  less  deformation,  or  a  greater  mod- 
ulus of  elasticity  than  those  which  were  not  reinforced,  just  as  was  noted  with 
regard  to  the  ordinary  (simply  longitudinally  reinforced)  concrete  prisms  already 
described. 

In  all  cases  the  load  was  noted  at  which  the  first  cracks  were  observed,  as 
well  as  the  ultimate  load.  The  cracks  appeared  first  in  the  concrete  layer  out- 
side the  spiral,  and  large  fragments  of  that  shell  finally  became  detached.  The 
types  of  failure  are  shown  in  Figs.  72  and  73. 

*  The  tests  were  made  at  the  Testing  Laboratory  of  the  Royal  Technical  High  School  at 
Stuttgart.  The  results  were  published  by  the  president,  Bach,  in  "  Druckversuche  mit  Eisen.- 
betonkorpen,  Versuch.  B."  Berlin,  1905.  Also  in  No.  29  of  " Mitteilungen  iiber  Forschungn- 
arbeiten." 


70  CONCRETE-STEEL  CONSTRUCTION 

Table  XXVII  gives  the  results  of  the  tests,  together  with  the  increase  in  strength 
developed  by  the  reinforced  specimens  as  compared  with  specimen  I,  which 
were  without  reinforcement. 

Table  XXVII 


RESULTS  OF  CONSIDERE  COLUMN  TESTS 


No. 

Figure 

Unit  Stress  of 
First  Crack 

Oi 

Increase  over 
Non-reinforced 
Column 
Oi—  133.  etc. 

Ultimate 
Unit  Load 

Increase  over 
Non-reinforced 
Column 
i33>  etc. 

Ultimate 
Unit  Load 
on  Central  Core 

kg/cm  2 

lbs/in^ 

kg/cm2 

lbs /in  2 

kg/cm2 

lbs/in2 

kg/cm2 

lbs/in^ 

kg/cm2 

lbs./in2 

I 

68 

133 

1892 

133 

1892 

II 

69 

159 

2262 

26 

370 

159 

2262 

26 

370 

230 

3272 

III 

69 

161 

2290 

28 

398 

178 

2532 

45 

640 

257 

3656 

IV 

69 

170 

2418 

37 

516 

240 

3414 

107 

1522 

347 

4936 

V 

70 

224 

3186 

91 

1294 

226 

3215 

93 

1323 

327 

4651 

VI 

70 

230 

3272 

97 

1380 

230 

3271 

97 

1379 

332 

4722 

VII 

70 

243 

3442 

110 

1550 

281 

3997 

148 

2105 

406 

5775 

VIII 

69 

196 

2788 

63 

896 

200 

2845 

67 

953 

289 

4111 

IX 

69 

170 

2418 

37 

526 

211 

3001 

78 

1 109 

305 

4936 

X 

69 

180 

2560 

47 

668 

256 

3641 

123 

1749 

370 

5263 

'XI 

69 

158 

2247 

25 

355 

246 

3499 

113 

1607 

355 

5050 

XII 

70 

163 

2318 

30 

426 

163 

2318 

30 

426 

236 

3357 

XII 

70 

164 

2333 

31 

441 

230 

3271 

97 

1379 

332 

4722 

XII 

70 

184 

2617 

51 

725 

302 

4295 

169 

2403 

436 

6202 

xiir 

71 

162 

2304 

29 

412 

162 

2304 

29 

412 

234 

3328 

XIII" 

71 

179 

2546 

46 

654 

181 

2574 

48 

682 

261 

3713 

XIIU" 

71 

186 

2646 

53 

754 

199 

2830 

66 

938 

298 

4239 

'  XIV ' 

71 

2205 

22 

313 

155 

2205 

22 

313 

224  . 

3185 

XIV" 

71 

183 

2603 

50 

711 

183 

2603 

50 

711 

264 

3755 

XIV"' 

71 

207 

2944 

74 

1052 

207 

2944 

74 

1052 

299 

4253 

Table  XXVIII  gives  the  results  of  applying  Considere's  formula  to  specimen? 
V,  VI,  and  VII,  with  ^^  =  133  (1888  lbs/in2)  and  (7,  =  2400  (3414  lbs/in2). 

Table  XXVIII 


COMPARISON  OF  RESULTS  OF  CONSIDERE  COLUMN  TESTS 


Concrete  Core 

Area  of  Reinforcement 

Strength  of 
Reinforcement, 

Ultimate  Strength 

No. 

Area. 

Strength, 
1.5X133^^ 
i.5Xi89oF^ 

Longitu- 
dinal 

(Spiral) 
Equivalent 
Longitu- 
dinal. 

2400 

(/„4-2.4/;) 
34140" 
(/^4-2.4/;) 

Kg. 

Lbs. 

cm2 

in2 

kg. 

lbs. 

cm2 

in2 

cm2 

in2 

kg. 

lbs. 

Ob- 
served 

Com- 
puted 

Ob- 
served 

Com- 
puted 

V 
VI 
VII 

452 
442 
432 

70.1 
68.5 
67.0 

90100 
88200 
86200 

198220 
194040 
189640 

7.60 
7.60 
7.60 

1. 18 
1. 18 
1. 18 

3-90 
7.78 
13-49 

0.60 
I  .  21 
2.08 

40700 
63000 

95900 

89540 
138600 
210980 

130800 
1 51 200 
182100 

142000 
144000 
176200 

287760 
332640 
400620 

312400 
3 I 6800 
387640 

COMPRESSION 


71 


In  spite  of  the  defects  in  the  specimens,  the  strength  developed  l^y  them 
corresponded  approximately  with  the  results  indicated  by  the  formula,  and 
exceeds  them  for  the  specimens  with  the  least  reinforcement. 

Considere  suggests  the  following  lessons  from  the  other  results. 

Pitch  of  the  Spirals. 

Specimens  XIII  and  XIV,  in  which  the  pitch  was  exaggerated  (80  and  120 
mm.  =3.15  and  4.72  in.),  gave  mediocre  results. 

Specimen  XIII''',  although  showing  an  increase,  did  not  develop  the  strength 
indicated  by  the  formula.  This  circumstance  appears  to  be  ascribable  to  a 
wrong  relationship  between  the  diameters  of  the  longitudinal  rods  and  of  the 
spiral  reinforcement.  This  deficiency  was  not  eliminated  by  a  decrease  in  the 
pitch  of  the  spirals.* 


Fig.  72. — Failure  from  shear.  Fig.  73. — Spalling  of  the  outer 

(Spiral  broken.)  concrete  shell. 

Relationship  Between  the  Spirals  and  the  Longitudinal  Rods. 

In  specimens  II,  III,  IV,  VIII,  IX,  X,  XI,  and  XII'  the  sectional  area  of 
the  longitudinal  rods  was  small,  and  the  results  were  consequently  indifferent; 
but  the  greater  the  total  weight  of  spiral  reinforcement,  the  higher  were  the 
results. 

On  the  whole,  the  tests  seem  to  prove  that  when  the  spirals  are  increased  in 
strength,  their  pitch  must  be  decreased,  and  the  cross-section  or  number  of  the 
longitudinal  rods  must  be  increased ;  "j"  for  with  increase  in  strength  of  spirals, 

*  Specimens  XIII  and  XIV  gave  practically  identical  results.  With  the  spiral  reinforce- 
ment diameter  and  pitcb  as  designed,  longitudinal  rods  of  larger  diameter  should  apparently 
have  been  used  in  XIII"  and  XIV"  to  give  results  proportional  to  the  corresponding  tests 
of  XII  and  XIV.— Trans. 

fin  order  to  secure  a  consistent  increase  in  supporting  power. — Trans. 


72 


CONCRETE-STEEL  CONSTRUCTION 


the  concrete  is  in  a  condition  to  resist  a  heavier  pressure  and  its  tendency  to 
•force  its  way  out  between  the  longitudinal  rods  also  increases. 

In  planning  the  programme  of  tests  of  hooped  concrete,  a  direct  comparison 
was  sought  with  the  column  tests  conducted  for  the  Reinforced  Concrete  Com- 
mission of  the  Jubilaumsstiftung  der  Deutscher  Industrie  (page  6i) ,  by  making 
the  cross-section  of  the  octagon  equal  a  square  area  25  by  25  cm.  (10  by  10 
in.  approx.),  and  by  so  arranging  the  spirals  and  longitudinal  rods  that  in  speci- 
mens II,  III,  IV,  V,  VI,  and  VII  the  amount  of  steel  in  the  spirals  was  equal  to 
that  of  the  ties  in  the  columns  reinforced  with  4  rods  15  mm.  (0.59  in.)  in 
diameter,  and  with  spacings  of  25  cm.  (9.8  in.)  12.5  cm.  (4.9  in.)  and  6.25  cm. 
(2.45  in.). 

While  with  the  ordinary  form  of  tie  the  increase  in  strength  compared  with 
non-reinforced  concrete  prisms  (3  months  old)  amounted  to  27  kg/cm^  (384 
lbs/in^),  36  kg/cm2  (512  lbs/in^),  and  64  kg/cm^  (910  lbs/in^);  with  the  employ- 
ment of  the  same  amount  of  steel  in  the  form  of  spirals,  and  with  4  longitudinal 
rods  only  7  mm.  (0.28  in.)  in  diameter  (5  to  6  months  old)  the  increase  was  26 
kg/cm^  (370  lbs/in^),  45  kg/cm^  (640  lbs/in^),  and  107  kg/cm^  (1520  lbs/in^); 
and  with  eight  rods  11  mm.  (0.43  in.)  in  diameter,  it  was  93  kg/cm^  (1320 
lbs/in^),  97  kg/cm^  (1380  lbs/in^),  148  kg/cm^  (2100  lbs/in^). 

In  the  last  instance  it  is  to  be  noted  that  the  eight  rods  of  11  mm.  (0.43  in.) 
diameter,  have  almost  exactly  the  same  section  as  the  four  15  mm.  (0.59  in.) 
rods  in  the  last  column  test,  so  that  the  advantage  of  the  spirals  over  the  hooped 
results  is  an  increase  of  strength  of  66  kg/cm^  (939  lbs/in^),  61  kg/cm^  (868 
lbs/in^),  and  84  kg/cm^  (1195  lbs/in^). 

In  the  prisms  VIII,  IX,  X,  and  XI  the  spirals  were  so  designed  that  the  quan- 
tity of  steel  in  them  was  equal  to  that  of  both  the  ties  and  longitudinal  rods  of 
the  column  tests  (page  61),  and  also  so  that  a  pitch  of  about  one-seventh  of 
the  column  diameter  was  obtained.  In  addition,  for  practical  reasons,  the  spirals 
were  held  in  position  by  four  longitudinal  rods,  7  mm.  (0.28  in.)  in  diameter.  The 
columns  with  four  rods  20  mm.  (0.79  in.)  in  diameter,  and  a  25  cm.  (9.8  in.)  spac- 
ing of  ties,  then  had  almost  exactly  the  same  amount  of  reinforcement  as  did  the 
spirally  reinforced  prism  IX,  and  also  as  did  the  column  with  four  rods  15  mm. 
(0.59  in.)  in  diameter  with  a  tie  spacing  of  12.5  cm.  (4.9  in.).  The  increase  in 
strength  of  the  ordinary  longitudinally  reinforced  columns,  as  compared  with 
the  non-reinforced  specimens,  according  to  Table  XXIV,  on  page  63,  amounted  to 

27  36  64  29  49  kg/cm^ 

(384)       (512)       (910)       (412)       (796  lbs/in2), 

and  in  the  case  of  prisms  VTII,  IX,  X,  IX  (again)  and  XI,  according  to  Table 
XXVII,  on  page  70,  it  amounted  to 

67  78  123  78  113  kg/cm2 

(952)      (iiio)       (1745)       (iiio)       (1605  lbs/in2). 

If  the  ratio  of  the  increase  in  strength  shown  by  these  two  series  of  tests 
representing  the  two  types  of  design,  but  employing  for  closer  comparison  only 


COMPRESSION 


73 


those  columns  of  the  one  kind  which  had  the  spacing  of  25  cm.  (9.8  in.),  is  com- 
puted, there  is  obtained 

67         Q     78       .  113 

—  =  2.48,    —  =  2.60,  ■ — -  =  2.21. 

27       ^  '    29        ^'  49 

These  results  are  in  satisfactory  agreement  with  the  figure  2.4  assumed  l)y 
Considere,  which,  therefore,  expresses  the  superiority  of  reinforcement  in  the 
form  of  spirals  over  its  value  in  the  form  of  longitudinal  rods. 

The  results  obtained  from  specimens  XII  disclose  the  importance  of  com- 
bining with  any  spiral  reinforcement  a  longitudinal  reinforcement  of  about  the 
same  proportions. 


CHAPTER  VI 


THEORY  OF  REINFORCED  CONCRETE 
SIMPLE  BENDING 

In  homogeneous  bodies  possessing  a  constant  modulus  of  deformation, 
equations  of  flexure  can  be  derived  on  the  assumption  that  sections  which  were 
plane  before  bending  will  be  plane  after  bending.  The  question  at  once  arises 
to  what  extent  this  assumption  will  apply  to  reinforced  concrete  bodies. 

By  experiments  with  homogeneous  bodies  of  rectangular  cross-section,  the 
correctness  of  this  assumption  has  been  established  within  certain  limits,  but 
it  owes  its  general  acceptance  to  a  demand  for  the  greatest  possible  simplification 
of  methods  of  calculation.  Furthermore,  it  is  known  that  this  assumption  of 
the  conservation  of  plane  sections  is  irreconcilable  with  the  existence  of  shearing 
stresses,  which  generally  tend  to  produce  an  S-shaped  deformation  of  any  right 
section.  With  equal  reason,  therefore,  there  can  be  assumed  the  conservation 
of  plane  sections  in  the  flexure  of  reinforced  concrete  beams,  and  it  is  to  be  noted 
that  the  strength  of  the  beam  computed  on  page  29,  on  the  basis  of  such  plane 
sections,  but  constructed  of  a  material  possessing  a  variable  modulus  of  defor- 
mation, coincides  satisfactorily  with  that  obtained  by  experiment. 

If,  therefore,  in  Fig.  74,  AB  represents  the  cross-section  of  a  reinforced  con- 
crete beam,  A'B'  is  the  curve  of  strain 
  and  the  corresponding  stress  curve  is  repre- 
sented by  EOF.    The  latter  is  really  a 
properly   plotted    stress-strain    curve  for 
concrete.     The   steel   reinforcement  must 
follow   the   deformation   of   the  concrete. 
The  upper   reinforcement  is  consequently 
I f   -^  shortened  an  amount  CC'^  while  the  lower 
TJ3' £                           layer  of  steel  is  stretched  an  amount  DD' . 
Fig.  74.  The  corresponding  steel  stresses  are  pro- 
portional to  these  strains. 
The  distribution  of  stress  in  reinforced  concrete,  shown  in  Fig.  74,  will  occur 
only  under  very  moderate  loading,  because  the  elasticity  of  concrete  in  tension 
is  soon  overcome.    This  condition  of  stress  is  designated  Stage  1.    In  calculat- 
ing stresses  in  this  stage,  the  lines  OE  and  OF  may  be  regarded  as  straight. 

With  increasing  load,  the  full  tensile  strength  of  the  concrete  will  be  attained 
throughout  the  whole  zone  of  tension,  and  if  the  great  capacity  of  reinforced 

74 


SIMPLE  BENDING 


75 


concrete  to  stretch,  observed  by  Considere,  is  assumed  provisionally,  then  the 
distribution  of  stress  under  such  conditions  will  resemble  Fig.  75.  This  may 
be  designated  Stage  II.  According  to  Considere's  tests,  this  second  stage  does 
not  extend  beyond  the  stress  in  the  concrete  corresponding  to  the  strain  of  the 
reinforcement  at  its  elastic  limit.  A  continued  increase  in  the  load  causes  the 
elastic  limit  of  the  steel  to  be  exceeded,  the  tensile  strength  of  the  concrete  is 
no  longer  a  factor,  and  finally  a  break  occurs  through  a  failure  in  the  tensile 
strength  of  the  steel  or  in  the  compressive  strength  of  the  concrete.  This  last 
condition  of  loading,  the  breaking  stage,  is  designated  Stage  III.  It  is  evident 
that  in  an  exact  theoretical  study  of  the  breaking  stage,  great  difficulties  are 
encountered;  since,  then,  the  elastic  conditions  usually  employed  as  the  basis  for 
calculations  do  not  exist. 

With  regard  to  Stage  II,  it  must  be  noted  that  no  certain  dependence  can 
be  placed  on  the  tensile  strength  of  the  concrete,  partly  because  of  irregularities 
in  its  composition,  but  especially  because  recent  tests  have  shown  that  the  stretch 
of  concrete  does  not  extend  nearly  to  the  strain 
of  the  steel  at  its  elastic  limit.    Cracks  in  the 
concrete  may  therefore  be  expected  in  the  latter 
part  of  Stage  II.    The  early  part  of  this  condi- 
tion of  loading   (without  tension  cracks  in  the 
concrete)   may  be  called  Stage  lla,  while  the 
latter  part  may  be  Stage  Wh  (with  tensile  cracks 
in  the  concrete  and  steel  stress  less  than  the 

elastic  limit).    The  distribution  of  stress  in  these         yig  75.  Fig" 76. 

two  sub-stages  is  shown  in  Figs.  75  and  76. 

The  question  arises,  which  stress  condition  is  to  be  made  the  basis  from 
which  to  derive  methods  of  calculation  for  practical  purposes.  When  consid- 
eration is  given  to  the  fact  that  the  object  of  every  static  calculation  is  not  so 
much  to  ascertain  the  exact  stresses  in  any  structure  resulting  from  a  given  load, 
but  rather  to  secure  an  adequate  degree  of  safety  for  the  structure,  then  it  must 
be  concluded  that  attention  should  be  given  to  the  examination  of  the  supporting 
power  of  reinforced  concrete  construction  subject  to  bending — in  Stage  III 
(that  of  failure).  This,  however,  can  hardly  be  accomplished  theoretically. 
Stage  I  must  be  excluded  from  consideration  because  it  has  already  been  passed 
even  with  perfectly  safe  loads.  Stage  \\a,  which  has  been  recommended  in 
several  instances  as  a  basis  for  calculations,  should  be  excluded  because  of  the 
uncertainty  of  Considere's  tests,  and  also  because  the  concrete  has  been  shown 
to  be  subject  to  cracks,  attributable,  variously,  to  deficient  manipulation,  inter- 
ruptions during  the  concreting  process,  to  effects  of  temperature  change,  or  to 
excessively  rapid  drying.  Further,  no  more  exact  information  is  obtainable 
from  this  stage  concerning  the  necessary  amount  of  steel  than  is  obtainable  from 
Stages  l\h  or  III. 

Stage  lib  is  thus  shown  to  be  the  only  stress  condition  readily  available  for 
theoretical  treatment  and  the  one  which  most  clearly  shows  the  required  amount 
of  reinforcement.  Moreover,  the  method  derived  from  it  has  the  great  advan- 
tage of  simplicity,  and  it  can  be  adapted  to  Stage  III,  so  that  safe  stresses  can 
be  selected  which  bear  a  proper  relation  to  the  results  of  ultimate  bending  tests. 


76 


CONCRETE-STEEL  CONSTRUCTION 


In  making  designs  based  on  Stage  11^  instead  of  on  the  rupture  stage,  no 
greater  error  is  committed  than  is  made  in  every  calculation  of  ordinary  timber 
or  steel  construction,  in  which  Navier's  theory  of  flexure  is  almost  always  employed, 
even  though  it  is  not  applicable  at  the  point  of  rupture. 

There  follow  a  few  methods  of  calculating  reinforced  concrete  structures 
subjected  to  bending  stress.  They  have  been  developed  on  the  basis  of  Stage  lib. 
In  them  it  is  assumed  that,  after  deformation,  the  strained  steel  sections  remain 
in  the  same  planes  with  the  corresponding  compressed  concrete  sections.  The 
tensile  strength  of  the  concrete  is  consequently  ignored. 

Rectangular  Sections — Slabs 

I.  With  rectangular  cross-sections  the  calculations  can  be  made  with  the 
aid  of  the  stress-strain  curve  exactly  as  has  already  been  described  for  non- 
reinforced  concrete  beams. 

In  Fig.  77  the  line  OE  represents  the  variation  in  compressive  stress.  The 
branch  for  tensile  stresses  is  omitted,  and  its  place  is  taken  by  the  tension  sur- 


FiG.  77. 


face  of  the  reinforcement.  Calling  the  width  of  cross-section  equal  to  unity, 
and  assuming  ob  and  Og  definite  safe  stresses  for  the  concrete  and  the  steel 
respectively,  then  the  stress  surface  for  the  concrete  is  definitely  determined, 
and  also  the  position  of  the  reinforcement.  The  tension  surface  of  the  steel  is 
a  long,  narrow  rectangle.  In  simple  flexure,  there  are  present  no  exterior  longi- 
tudinal force  components,  and  consequently  the  tensile  and  compressive  forces 
must  balance  in  each  section.  Or,  the  area  of  the  compression  surface  must  be 
equal  to  the  rectangle  of  the  tensile  stress.  If  the  distance  of  the  centroid  of 
the  reinforcement  from  the  upper  edge  is  called  h,  then  fe  can  be  expressed  as 
a  function  of  h,  ob,  and  the  moment  M.  The  latter  is  equal  to  the  area  of  the 
compression  surface  multiplied  by  the  distance  of  its  centroid  from  the  rein- 


SIMPLE  BENDING 


77 


forcement,  and  consequently  fe  will  be  a  function  of  h'^;  h  may  be  called  the 
useful  height  of  section.  Determining  dimensions  of  members  according  to  this 
method  is  easy,  whereas  complicated  trial  calculations  are  necessary  to  ascertain 
the  stresses  in  an  existing  structure. 

This  method,  in  connection  with  the  stress-strain  curves  shown  on  page  24, 
gives  the  following  results:  With  a  1:4  mixture  with  14  per  cent  of  water, 
if  Fe  represents  the  section  of  the  reinforcement  in  square  centimeters  per 
meter  width  of  slab  (also  square  inches  per  foot  width),  and  M  is  computed 
for  the  same  width  in  centimeter-kilograms  (inch-pounds)  with  (76=40  kg/cm^ 
(569  lbs/in^),  cre  =  1000  kg/cm^  (14,220  lbs/in^),  £e  =  2,160,000  kg/cm^  (30,720,000 
lbs/in2). 

h=o.o^o'i\/ M  centimeters  per  meter  width  of  M; 

*(/7.=o.o3i2\/il/  inches  per  ft.  width  of  M); 

Fe=o.o277\/i/  square  centimeters  per  meter  width; 

*(Fe=o.oo255V^M  square  inches  per  ft.  width). 

The  thickness  d  of  the  slab  is  to  be  taken  1.5  to  2.0  cm.  (0.6  to  0.8  in.)  greater 
than  the  calculated  useful  height  h,  the 


bottom  face  of  the  concrete  being  lowered 
to  that  extent. 

It  is  further  to  be  noted  that  this  method 
also  allows  account  to  be  taken  of  the 
tensile  strength  of  the  concrete. 

2.  The  same  general  method  may  be 
followed  along  strictly  analytical  lines,  by 
employing  the  exponential  law.  The 
conservation    of    plane    sections    is  ex- 


pressed in  the  nomenclature  of  Fig.  78,  by  Fig.  78. 

the  proportion 

a~  h' 

whence 

£b  =  T  Se.     Also  ee  =  -^. 

0  rL 
Now,  according  to  the  exponential  law 

e.b=a  ob^, 

*  These  values  are  for  unit  stresses  of  569  and  14,220  lbs.  in  concrete  and  steel  respectively, 
and  for  a  moment  equal  to  M  in-lbs.  per  foot  width  in  English  units,  while  the  formula  in 
metric  units  is  for  M  kg-cm  per  meter  width. — ^Teans. 


78  CONCRETE-STEEL  CONSTRUCTION 

whence 

wherein  E  represents  the  modulus  of  elasticity  of  steel. 
Now, 


b  =  h—a, 


from  which  (h—a)  a  ob^=a-^, 


,  ha  Gh"^ 
whence  a  =  


E 


The  moment  M  is: 


Now,  (7^:(T6^:  :7^:a, 

so  that 


and  from  differentiation 


a  a' 

v=^  


dv=  o^~^d(7. 


Therefore, 


M=  {  a^-^dG[b-\  

Jo     ob^  \  o,^l 

(76"*  Jo  Ob-'^Jo 


=  a  0  ob-rd'^  \ —  Ob. 

m  +  i  2m  +  i 


After  substituting  the  values  of  a  and  6,  given  above, 
m  (76"*+^  E 


m  +  i  /i 


(^-^e  +  Ob'^E^' 


/(m+i)  (76"*  £ ^ \ 


SIMPLE  BENDING 


79 


The  area  of  steel,  }e,  for  unit  width  of  slab,  is  given  by  the  equation 


a  dv 


I    r"b  (7  a  m  a 


-da 


a  m  Ob 


m+l 


Oe  m  +  I  Ob' 


a  Ob 


Oe 


If  h  from  the  equation  for  M  is  substituted  herein,  there  results  for  unit  width 
of  slab 


Ob 


Oe 


 —  obE^ — ob"^ 

2m  + 1  « 


If  values  of  ob  and  Oe  are  assumed,  the  equations  given  above  may  be  employed 


Ob 


1-17 


E 


in  designing  slabs.    In  this  case  with  a=  ,  w=i.i7,  Sb-- 

230,000  230,000' 

2,160,000,  (76  =  40  kg/cm^,  and   ^7^=1000  kg/cm^,  if  Fe  represents  the  section 

of  the  reinforcement  for  one  meter  breadth  of  slab,  and  further  if  M  is  calculated 

for  the  same  breadth  in  centimeter-kilograms. 

/z=o.o363\/ M  centimeters  for  M  on  meter  width; 
*(/z=o.o2  78\/il/  inches  for  M  on  foot  width); 
Fe=o.o324\/ilf  square  centimeters  for  M  on  meter  width; 
*(Fe  =  o.oo298\/ M  square  inches  for  M  on  ft.  width). 

The  thickness  of  the  slab  is  to  be  increased  1.5  to  2  cm.  (0.6  to  0.8  in.)  over 
the  value  of  //,  this  increase  being  made 
below  the  centroid  of  the  reinforcement. 

3.  While  the  two  methods  above  de- 
scribed permit  only  of  designing — which 
is  most  important  for  the  engineer — 
the  following  method  may  be  employed  to 
investigate  the  stresses  in  completed  or 
completely  designed  reinforced  concrete 
slabs.    It  is  contained  in  the  "  Leitsatze"  ig.  79. 

of  the  Verbands  Deutsche  Architekten-  und  Ingenieurvereine,  and  the  Deutschen 


!      '       !  ^ 

1  1 

!      1  ^ 

*  See  foot-note,  page  77. 


80 


CONCRETE-STEEL  CONSTRUCTION 


Betonvereins,  of  1904,  and  is  also  included  in  the  Prussian  Ministerial  Regu- 
lations "  of  1904  and  1907. 

Here,  instead  of  the  exponential  law,  is  employed  the  proportionality  between 
the  tensile  and  compressive  stresses  of  the  concrete,  while  the  tensile  strength 
is  again  ignored.  If  a  constant  modulus  of  elasticity  of  concrete  Eh  is  assumed, 
and  the  distance  of  the  neutral  plane  from  the  top  of  the  slab  is  called  x,  then 
(see  Fig.  79), 


If  h  represents  the  assumed  breadth  of  the  section,  and  Fe  the  total  area  of 
the  tension  reinforcement  in  the  same  breadth,  then 


Z=D 


Further,  the  strains  are  in  the  proportion 


Ob  Ge    ,j  . 


whence 


2  Ge  F( 


hx 


Substituting  this  value  in  the  proportion  above,  gives 


2  Oe  Fe 


\x=—:{h—x) 


bxEb 


whence 


_2Fe 

Ee    b  Eh 


(h-x). 


With  this  may  be  transformed  into  the  quadratic  equation 

c,  .      Fe  1 

+  2  —  n  x  =  2  —  n  fi, 


from  which 


With  the  value  of  x  from  this  equation,  there  may  be  found 


SIMPLE  BENDING 


81 


and  the  maximum  stress  on  the  concrete 


_2D_  2M 

and  on  the  steel 


Z  M 


The  position  of  the  neutral  axis  is  determined  by  the  condition  that  it  must 
pass  through  the  centroid  of  the  effective  stress  surface,  in  which  the  area  of  the 
reinforcement  has  been  replaced  by  a  concrete  area  w-times  larger  than  that  of  the 
steel.  The  neutral  axis  thus  forms  the  lower  lim-it  of  the  compressed  portion 
of  the  concrete  section. 

This  gives  as  the  equation  for  the  statical  moment  of  the  effective  areas  with 
respect  to  the  neutral  axis 

X 

bx  n  Fe(h—x)  =o, 


from  which  follows  the  quadratic  equation 

r,  .      Fe  J 


A  formula  may  also  be  deduced  in  terms  of  the  unit  stresses. 
From  the  proportion 

Oh  . 

there  follows 

h  Ob  n 


X  =  - 

oe-\-n  Oh 

The  moment  M  for  breadth  h  is 
h  obx 


h  oh^  n  /  h  obn  \ 
■e+nob)\  3{oe+nob)/ 

{30e  +  2nOb). 


_b_ 

2{Oe 

h     ob^  n 


6{oe-\-n  Ob) 


82 


CONCRETE-STEEL  CONSTRUCTION 


The  total  area  of  steel  for  breadth  b  is 


(j;,  X  b 

re  =  , 

or 

b  h  oh^  n 


2  ae{oe-\-nab) 


If  the  safe  unit  stresses  adopted  in  the  "  Leitsatze  "  are  employed,  (7^=40 
kg/cm2  (569  lbs/in^),  (7^  =  1000  kg/ cm^  (14,220  lbs/in^),  if  also  6  =  100  cm.  (39.4 
in.)  and  ^  =  15,  there  results 

/j=o.039oV M  centimeters; 
Fe=o.0293\/ M  square  centimeters; 
*(/^=o.o2993\/ M  inch  for  one  foot  width  of  M)] 
*(Fe  =  0.002696V M  square  inches  for  one  foot  width  of  M). 

When  employing  the  safe  stresses  (75=40,  and  (7e  =  iooo,  the  area  of  reinforce- 
ment bears  to  h  the  ratio 

^  0.02Q^- 

Fe=  —h=o.n ^oh  (metric) ; 

0.0390  ^  ^' 

0.002696 

re=  — /j=o.oq/^  (hnglish). 

0.02993 


If  this  ratio  is  exceeded,  the  steel  cannot  be  fully  utilized,  because  then  the 
concrete  would  be  over-stressed  in  compression.  Reinforcement  of  this  character 
would  therefore  be  impracticable. 

With  variously  assumed  values  of  ob  and  Oe  the  distance  x  of  the  neutral  axis 
from  the  upper  edge  of  section  may  be  expressed  in  terms  of  h,  and  with  n  =  i^y 
and  6  =  100  cm.,  the  values  in  Table  XXIX  are  obtained. 

The  figures  in  heavy  type  represent  stresses  adopted  in  the  Leitsatze  "  and 
the  Prussian  ''Regulations."  With  (76=40,  and  (7^  =  1000  kg/cm^  (569  and  14,220 
lbs/in^,  respectively),  the  neutral  axis  is  located  at  f  of  the  height,  and  the  arm 
of  the  couple  formed  by  the  tensile  and  compressive  stresses  is  |//.  These 
results  are  of  great  value  in  preliminary  calculations  and  rough  estimates,  since 
with  moderate  concrete  stresses  these  quantities  do  not  vary  much. 

If,  for  instance,  a  continuous  roof  slab  is  to  be  designed,  of  which  the  greatest 
moment  is  70,000  cm. -kg.  (60,630  in.-lbs.),  there  must  first  be  determined  the 
thickness  and  steel  section  at  points  of  maximum  moment  (by  means  of  Table 
XX,  for  instance) ,  and  then  the  section  of  steel  at  the  points  of  minimum  moment 
by  the  formula 

F-  ^ 
re  —  — =y. 

(7et^ 


*  See  foot-note, -page  77. 


SIMPLE  BENDING 


83 


Table  XXIX 
BEAM  ELEMENTS  FOR  VARIOUS  UNIT  STRESSES 


Oc 

hl\/M* 

Fels/M* 

x/h. 

\     3  /  / 

kg/cm^ 

Ibs/in2 

kg/cm  2 

Ibs/in2 

cm.  for  M 
per  m. 

in.  for  A/ 
per  ft. 

cm^  for  M 
per  m. 

in  2  for  M 
per  ft. 

30 

427 

750 

10673 

0.0451 

0.0348 

0.0338 

0.00314 

0.375 

0.875 

X  ^ 

00 

498 

1  J 

1067^ 

0.0401 

0.0^07 

0.0^8=; 

0.00354 

0.412 

0.863 

40 

c;6o 

10673 

0.0^6^ 

0.0270 
•     /  y 

0.0430 

0.00394 

0.444 

0.852 

640 

1067^ 

o.o;^4. 

0.0256 

0.0474 

O.OOd^2 

0.474 

0.842 

712 

750 

10673 

0.0310 

0.0238 

0.01^17 

0.00476 

0.500 

0-833 

30 

427 

800 

11376 

0.0459 

0-0353 

0.0309 

0.00284 

0.360 

0.880 

1  ^ 

0  J 

4Q8 

800 

1 1  ^76 

'■'-01 

0. 0408 

0  OXXA 

o.oo'?2=; 

0.396 

0.  oOo 

40 

c;6q 

800 

1 1376 

0.0367 

0.0282 

0.0^07 

o„oo^67 

0.429 

0.857 

A^ 

640 

800 

11376 

0.0339 

0.0261 

0.0436 

0.00402 

0.458 

0-843 

712 

800 

11376 

0,0314 

0.0241 

0.0475 

0.004"?  7 

0.404 

0.839 

30 

427 

900 

12798 

0.0474 

0-0364 

0.0264 

0.00243 

°=333 

0.009 

J  J 

408 
ty>^ 

000 

y 

12798 

0.0420 

0.0^2^ 

0.0301 

0.00277 

O.3O0 

0.877 

40 

5q8 

900 

12798 

0.0380 

0. 0298 

0.0^^7 

0.00310 

0.400 

0.867 

4^^ 

640 

900 

12798 

0.0348 

0.0268 

o.o-?7^ 

0.00344 

0.429 

0-857 

50 

0 

712 

900 

12798 

0.0322 

0.0248 

0.0407 

0.00^,74 

^•455 

0.040 

20 

299 

1000 

14220 

0.0685 

0.0526 

0.0158 

0.00145 

0.230 

0.923 

J 

1000 

14220 

0.0568 

0.0436 

0.0193 

0.00178 

0.273 

0.  909 

30 

427 

1000 

14220 

0.0490 

0.0376 

0.0228 

0.00210 

0.310 

0.896 

35 

498 

1000 

14220 

0-0433 

O-O333 

0.0261 

0.00241 

0.344 

0.885 

40 

569 

1000 

14220 

0.0390 

0.0299 

0.0293 

0.00270 

0-375 

0.875 

45 

640 

1000 

14220 

0-0357 

0.0274 

0 . 03  24 

0.00301 

0.403 

0  866 

50 

712 

1000 

14220 

0-0330 

0.0253 

0.0354 

0.00326 

0.429 

0-857 

30 

427 

1200 

17076 

0.0519 

0.0398 

0.0177 

0.00164 

0.273 

0.909 

35 

49S 

1200 

17076 

0.0457 

0.0357 

0.0203 

o.ooiSS 

0.304 

0.898 

40 

569 

1200 

17076 

0.0410 

0.0315 

0.0228 

0.00210 

0-333 

0.889 

45 

640 

1200 

17076 

0.0375 

0.0288 

0-0253 

0.00234 

0.360 

0.880 

50 

712 

1200 

17076 

0.0345 

0.0255 

0.0277 

0.00255 

0-385 

0.872 

Exact  calculations  will  give  stresses  slightly  smaller  than  (7e  =  iooo  kg/cm^ 
(14,220  lbs/in^),  so  that  a  somewhat  greater  factor  of  safety  is  secured. 

If  an  existing  design  is  to  be  checked,  the  equations  of  page  80  must  be 
employed,  or  Table  XXX  used.  In  the  latter  case  it  is  only  necessary  to 
find  the  assumed  section  of  reinforcement  Fe  in  terms  of  the  useful  area  (for 

instance,  Fe  =  pLbh,  or  ^iid  then  the  values  of  x,  oi^,  and  Oe  may  be 

found  immediately. 

If  the  reinforcement  is  taken  at  approximately  0.79  per  cent  of  the  useful 
cross-section,  the  stress  of  the  extreme  layer  ob  will  be  equal  to  that  in  a  homo- 

M 


geneous  section,  i.e.,  oh  = 


A  value  of*  0.75  per  cent  also  approximates 


*  M  is  measured  in  kg.-cm.  in  one  column  and  in  in. -lbs.  in  the  other,  but  in  each  case  the 
■coefi&cients  are  computed  for  the  same  numerical  value  of  M. — Trans. 


84  CONCRETE-STEEL  CONSTRUCTION 

the  customary  amount  of  reinforcement  employed,  so  that  oh  may  be  computed 
in  this  simple  manner  with  sufficient  accuracy. 


Table  XXX 

BEAM  ELEMENTS  FOR  VARIOUS  PERCENTAGES  OF  REINFORCEMENT 


X 

Ob 

Oe 

Oe 

Per  Cent. 

T 

M/bh^ 

M/bh^ 

0.0100 

0  18 

b  009 

Tit 

116 

0.0095 

0*410 

c  6  cn 
5  -"bo 

o .  90 

0 . 0090 

0 . 402 

J  ■  /-I  / 

128 

0.85 

0.0085 

0.393 

5-852 

23.1 

135 

0.80 

0.0080 

0.  ^84 

5.968 

24.0 

0.75 

0.0075 

0-375 

6.096 

25.0 

152 

0.70 

0.0070 

0.365 

6.236 

26.1 

163 

0.65 

0.0065 

0.355 

6.394 

27-3 

T74 

0.60 

0 . 0060 

0.344 

6.572 

28.6 

188 

0-55 

0.0055 

0.332 

6.774 

30.2 

204 

0.50 

0.0050 

0.320 

7.006 

32.0 

224 

0.45 

0.0045 

0.306 

7.278 

34.0 

247 

0.40 

0 . 0040 

0. 292 

7-597 

36-4 

277 

0.35 

0.0035 

0. 276 

7.985 

39-4 

315 

0.30 

0.0030 

0.258 

8.471 

43-1 

365 

0.25 

0.0025 

0.239 

9.096 

47-8 

435 

0. 20 

0.0020 

0. 217 

9-945 

54-2 

539 

The  values  of  h  and  Fe  for  various  moments  are  contained  in  Table  XXX. 

Commencing  on  page  88  are  to  be  found  some  examples  of  computations 
in  full  which  were  given  in  the  "  Leitsatze." 

In  the  "  Zentralblatt  der  Bauverwaltung  "  for  1886  is  to  be  found  an  approxi- 
mate rule  devised  by  Konen,  which  is  frequently  employed  in  determining  the 
necessary  section  of  reinforcement.  It  makes  the  inaccurate  assumption  that 
the  neutral  plane  is  at  the  center  of  the  slab,  and  that  the  distance  between  the 
centroids  of  the  compression  and  tension  areas  is  J^/,  so  that  the  area  of  steel 
is  given  by  the  formula 

M 

re  —  T-;. 

Oe  \  d 

The  distance      is  correct  in  accordance  with  what  is  shown  on  page  83,  if 

id=o.S'jsK 
or  d=lh. 

This  equation  will  usually  hold  for  slabs  of  thicknesses,  d=6  to  12  cm.  (2.4. 
to  4.7  ins.)  so  that  in  such  cases,  approximate  calculations  can  be  made  with; 
|J  in  place  of  \h. 

Concerning  tests  made  with  rectangular  slabs,  see  page  90. 


SIMPLE  BENDING 


85 


Table  XXXI 
BEAM  ELEMENTS  FOR  VARIOUS  MOMENTS 
(76  =  40  kg/cm^  (569  lbs/in^),     <7g  =  iooo  kg/cm^  (14220  lbs/in^) 


M  for  Meter  Width. 

h 

d 

F 

c 

Corre- 

sponding 

cm.  for  M 

in.  for  M' 

cm-  for  M 

in2  for  M' 

M'  per  ft. 
width. 

cm. -kg 

in. -lbs. 

per  meter 

per  foot 

cm. 

in. 

per  meter 

per  foot 

in. -lbs. 

width. 

width. 

width. 

width. 

1 0000 

8661 

3-90 

1-54 

5-0 

1-97 

2-93 

O-I39 

2640 

1 1 000 

9428 

4.09 

1.61 

5-0 

1.97 

3-07 

0.145 

2904 

1 2000 

10394 

4.27 

1.68 

5-5 

2.17 

3.20 

0.151 

3168 

13000 

1 1  260 

4-44 

1 .74 

5-5 

2.17 

3-33 

0.158 

3432 

14000 

I  21  26 

4.62 

1. 81 

6.0 

2.36 

3-46 

0. 164 

3696 

15000 

12992 

4.78 

1.87 

6.0 

2.36 

3-58 

0. 169 

3960 

16000 

4-94 

1-94 

6.0 

2.36 

3-70 

0-175 

4224 

17000 

14724 

5 -09 

2.02 

6-5 

2.65 

3.81 

o.iSi 

4488 

18000 

15590 

5-24 

2.08 

6-5 

2.65 

3-93 

0.186 

4752 

19000 

16456 

5.38 

2.13 

6.5 

2.65 

4-03 

0. 191 

5016 

20000 

I732I 

5-52 

2.17 

6.5 

2.65 

4.14 

0. 196 

5280 

22000 

19054 

5-72 

2.25 

7.0 

2.76 

4-30 

0. 202 

5808 

24000 

20786 

6.04 

2-38 

7.0 

2.76 

4-53 

0.215 

6336 

26000 

22518 

6.29 

2.48 

7-5 

2-95 

4.71 

0.223 

6864 

28000 

24251 

6.53 

2-57 

8.0 

3.15 

4.91 

0.233 

7392 

30000 

25984 

6-75 

2.66 

8.0 

3.15 

5.06 

0.240 

7920 

32000 

27716 

6.98 

2.76 

8-5 

5-35 

5-22 

0.247 

8448 

34000 

29448 

7. 20 

2.84 

8-5 

3-35 

5-39 

0.255 

8976 

36000 

31180 

7.40 

2.91 

8-5 

3-35 

5-54 

0. 262 

9504 

38000 

32913 

7.61 

3.00 

9.0 

3-54 

5-70 

0. 270 

10032 

40000 

34645 

7.80 

3-07 

9-0 

3-54 

5-85 

0. 277 

10560 

42000 

36377 

8.00 

3.15 

9.0 

3-54 

6.00 

0.284 

11088 

44000 

38109 

8.19 

3.23 

9.5 

3-74 

6.13 

0. 290 

I1616 

46000 

39832 

8.37 

3-30 

9-5 

3-74 

6.28 

0.297 

1  2144 

48000 

41574 

8.56 

3-37 

10.0 

3-94 

6.42 

0.304 

12672 

50000 

43307 

8-74 

3-44 

10.0 

3-94 

6-55 

0.310 

13200 

55000 

47637 

9-15 

3.60 

10.5 

4-13 

6.86 

0.324 

14520 

60000 

51968 

9-56 

3-76 

11.0 

4-33 

7.16 

0.339 

15840 

65000 

56298 

9-94 

3-91 

II-5 

4-54 

7-45 

0-352 

17160 

70000 

60630 

10.32 

4.06 

12.0 

4.72 

7-74 

0.366 

18480 

75000 

64959 

10.68 

4.19 

12.0 

4.72 

8.01 

0-379 

19800 

80000 

69291 

11 .05 

4-34 

12.5 

4.92 

8.29 

0-392 

21  I  20 

85000 

73620 

11.38 

4.46 

12.5 

4.92 

8.53 

0.403 

22440 

90000 

77952 

11 . 70 

4.60 

13.0 

5.12 

8-75 

0.414 

23760 

95000 

82282 

12.04 

4-74 

13.5 

5.72 

9-03 

0.427 

25080 

I 00000 

86614 

4.85 

1^.5 

5.72 

9.27 

0.438 

26400 

105000 

90944 

12.67 

4-97 

14.0 

5-51 

9-50 

0.449 

27720 

I I 0000 

94280 

12.90 

5.07 

14.0 

5-51 

9.68 

0 . 459 

29040 

1 1 5000 

98611 

13-23 

5.21 

14-5 

5-71 

9.92 

0.469 

30360 

1 20000 

103940 

13-52 

5.32 

15.0 

5-90 

10. 14 

0.479 

31680 

125000 

108270 

13.80 

5-43 

15-5 

6.18 

10.35 

0.489 

33000 

130000 

1 1 2600 

14.05 

5-53 

15.5 

6.18 

10.54 

0.498 

34320 

135000 

116930 

14.33 

5-64 

16.0 

6.30 

10.75 

0.508 

35640 

140000 

121260 

14.60 

5-75 

16.0 

6.30 

10.95 

0.518 

36960 

145000 

I2559I 

14.87 

5-85 

16.5 

6.49 

11.15 

0.528 

38280 

86 


CONCRETE  STEEL  CONSTRUCTION 


Table  XXXI — Continued 


(76  =  40  kg/cm^  (56.9  lbs/in2),     (7e  =  iooo  kg/cm^  (14220  lbs/in^) 


M  for  Meter  Width. 

h 

d 

A 

Corre- 

sponding 

cm.  for  M 

in.  for  M' 

cm2  for  M 

in  2  for  M' 

Ivl  XL* 

width. 

cm.  =k. 

in. -lbs. 

per  meter 

per  foot 

cm. 

in. 

per  meter 

per  foot 

in. -lbs. 

width. 

width. 

width. 

I  50000 

129920 

15-13 

5-96 

10.5 

0.49 

1^-35 

O-530 

39600 

160000 

138580 

15.60 

6. 14 

17.0 

0 . 09 

1 1 . 70 

0.554 

42240 

1 70000 

147240 

16. 10 

6  Id. 

18.0 

7.09 

12.07 

0.571 

44880 

180000 

155900 

16.60 

6.  54 

18.5 

7.29 

12.45 

0-589 

47520 

190000 

164560 

17.00 

6.69 

19.0 

7.4« 

12.75 

0.603 

50160 

200000 

173210 

17.45 

6.87 

19-5 

7 .  oo 

13.09 

0.619 

52800 

210000 

181870 

17.87 

7-04 

20 . 0 

7.»7 

13-45 

0.030 

55440 

220000 

190540 

18.30 

7  .  21 

20.5 

8.07 

13-74 

0.649 

58080 

230000 

199200 

18.71 

7.  ^7 

21.0 

8.27 

14.06 

0.664 

60720 

240000 

207860 

19.12 

7.  =^3 

/  -  JO 

21-5 

14-35 

0.  070 

63360 

250000 

216520 

19-50 

7.68 

22,0 

R  AA 

14-65 

0  .  692 

66000 

260000 

225180 

19.89 

7.83 

22.5 

0 .  ou 

14.95 

0.707 

00040 

270000 

2^^840 

20.  26 

7.98 
/  ■  y 

23.0 

9-05 

15-23 

0.  720 

71280 

280000 

242510 

20.64 

8.1^ 
'J 

23.0 

9-05 

15-51 

0.733 

73920 

290000 

251170 

21  .00 

8.27 

23-5 

9-25 

15-70 

0.  742 

76560 

300000 

259840 

21.36 

8.41 

24.0 

9-45 

16.05 

0.759 

79200 

320000 

277160 

22.06 

8.69 

24-5 

9-65 

10 .  50 

0.704 

Q  ^  ,1  R^ 
0440O 

340000 

294480 

22.  74 

8-95 

25.0 

9-84 

17.08 

0.807 

89760 

360000 

311 800 

2^  .AO 

0  21 

26.0 

10.24 

17.58 

0-831 

95040 

380000 

^201 ^0 

2  A.  OA 

Q  47 

26.5 

10.43 

iO  .  OU 

0.053 

100320 

400000 

346450 

24.67 

9.71 

27.0 

10.63 

18.54 

0.070 

105600 

420000 

363770 

25.27 

9-95 

2o.O 

1 1 .02 

Ib.99 

0.095 

I 10880 

440000 

381090 

2^  87 

10.19 

28.5 

11.22 

19.44 

0.919 

I16160 

460000 

398320 

26.45 

10.41 

29.0 

II  .42 

19.87 

0.939 

I  21440 

480000 

415740 

27.02 

10.64 

29-5 

II  .61 

20.30 

0.959 

I  26720 

500000 

433070 

27-58 

10.86 

30.0 

T  T  Rt 

20  .72 

0.979 

132000 

550000 

476370 

28.92 

11-39 

31-5 

12.  40 

21-73 

I  .027 

145200 

600000 

519680 

^0.21 

11.89 

33-0 

12.99 

22.70 

1-073 

I  58400 

650000 

562980 

31  -44 

12.38 

34-0 

13-38 

23.63 

I.  117 

I  71600 

700000 

606300 

■^2.64 

12.85 

35-0 

13-7^ 

24.52 

I  -159 

I04000 

750000 

649590 

33-76 

13.29 

36.5 

14-37 

25.39 

I  .  200 

198000 

800000 

692910 

34-88 

13-73 

37-5 

14.76 

26. 20 

1.238 

211 200 

850000 

736200 

0  J  '  y  3 

14..  I 

38.5 

15.16 

27.01 

I  .  276 

224400 

900000 

779520 

37-01 

14.57 

39-5 

15-55 

27.79 

I-3I3 

237600 

950000 

822820 

38.01 

14.96 

40.5 

15-94 

2^^.55 

1-349 

250800 

I 000000 

866140 

39.00 

15-35 

1";  53 

29.30 

^  -305 

I I 00000 

942800 

40.90 

16. 10 

43-5 

17.12 

30.62 

1.447 

290400 

1 200000 

1039400 

42.72 

16.82 

45-5 

17.91 

32.10 

I. 517 

316800 

1300000 

1 1  26000 

44.46 

17.50 

47-5 

18.70 

33-39 

1.578 

343200 

1400000 

I  21 2600 

46.14 

18.17 

49.0 

19.29 

34.65 

1.637 

369600 

1 500000 

1299200 

47-77 

18.78 

50.5 

19.88 

35-86 

1.695 

396000 

1600000 

1385800 

49-32 

19.42 

52.0 

20.47 

37.02 

1.749 

422400 

SIMPLE  BENDING 


87 


RECTANGULAR  SECTION,  DOUBLE  REINFORCEMENT 


Fig.  So. 


-J? 


Where  reinforcement  is  placed  within  the  zone  of  compression  but  is  of  such 
size  as  to  be  far  subordinate  in  effect  to 
the  concrete,  and  with  the  assumption  of  a 
constant  modulus  of  elasticity,  calculations 
may  be  carried  out  according  to  the  method 
which  follows  and  which  corresponds  with 
process  3,  last  preceding. 

With  the  nomenclature  of  Fig.  80  there 
is  obtained  for  simple  flexure,  from  the 
equality  of  tensile  and  compressive  stresses 
in  the  cross-section,  the  equation: 


 f_ 

_t.  > 

A 
1 

1 

1 
• 

Fe  Oe  =  ~Ob  X  +  FeOe'j 
2 


(I) 


wherein  the  small  reduction  in  the  area  of  the  concrete  by  the  steel  section  Fe  is 
ignored. 

Further,  the  following  relations  must  hold: 


Oh  ^  Oe 
Eb  '  Ee 


=x:{h—x)  (2) 


Ob  ^  Oe 
Eb'  Ee 

X  h 


x:{x-h'). 


Ob 


h-^j+Fe'Oe\h-h'). 


(3) 
(4) 


These  four  equations  suffice  for  the  determination  of  the  four  unknowns 

Ee 

<7e,  o/,  Of^,  if  the  remaining  quantities  are  known.  From  (2)  and  (3)  with  -Er=n 
there  results 

ob{h—x)n  ,  ^ 

Oe=  >  (5) 


,    ob(x  —  h')n 

Oe  =  . 

X 


(6) 


When  these  values  are  inserted  in  (i)  there  results  a  quadratic  equation  from 
which  X  may  be  derived 


„  Fe-\-Fe       271  ,  7  /  77  /\ 

x^  +  2xn  7  =  -rQi  Fe+h'Fe). 

0  0 


(7) 


The  same  value  may  be  deduced  from  the  condition  that  the  neutral  axis 
passes  through  the  centroid  of  the  effective  section,  in  which  the  area  of  steel  has 


88 


CONCRETE-STEEL  CONSTRUCTION 


been  replaced  by  an  equivalent  area  of  concrete  n  times  larger,  and  the  centroid 
at  the  same  time  lies  on  the  lower  edge  of  the  compressed  concrete  zone. 
From  the  solution  of  equation  (7) 


b       J    '  b 

With  X  determined,  (t^  is  obtained  from  equation  (4) 

"^^"bx^ish-x)  +6  Fe'  n{x-h'){h-hy 

and  Oe  and      are  given  by  equations  (5)  and  (6). 

If  Fe=o  in  equations  (7)  and  (8)  there  result  the  values  given  on  page  80 
for  single  reinforcement. 

Exactly  as  for  single  reinforcement,  condensed  formulas  for  quick  designing 
may  be  developed,  but  they  possess  no  practical  value. 

Example. — A  reinforced  concrete  slab  100  cm.  (39.4  in.)  wide  is  to  resist  a 
bending  moment  of  600,000  cm. -kg.  (519,680  in. -lbs.)  but  a  thickness  of  30  cm. 
(7.62  in.)  cannot  be  exceeded.  Since,  according  to  Table  XXXI,  a  thickness  of 
33  cm.  (8.38  in.)  is  necessary,  a  fiber  stress  cr^  more  than  40  kg/cm^  (569  lbs/in^) 
will  be  developed  if  none  but  lower  reinforcement  is  provided,  and  such  as  will 
make  (Te  =  iooo  kg/cm^  (14,223  lbs/in^).  In  fact,  with  ^^  =  28. 5  cm^  (4.42  in^) 
(7b=46.$  and  (Tg  =  ioio  kg/cm^  (661  and  14,365  lbs/in^,  respectively).    In  order 

^  f,^fQ(^^i..^  !-=  b'1.00  ^, 


>f — I  5^  rr: 


FT-  ^  •  i^e\  Sr 

d-'jo^  --T--  d^SO  \  

;  1  .  .^:f  U  I  .  ..ill'.  .  .  .  U.^^.- 

Fig.  81.  Fig.  82. 

to  reduce  Ob,  upper  reinforcement  is  introduced  amounting  to  ^^  =  9.5  cm^ 
(1.47  in2),  and  there  is  then  obtained  with  /^  =  27  cm.  and  h'  =  s  cm.  (10.63  and 
1. 1 8  ins.  respectively).    (See  Fig.  81.) 


100  ^  100^  100 


=  10.8  cm.  (4.75  ins.); 

6X600,000X10. 


100X10.8^(3X27 -10.8) +6X9.5  Xi5(io.8-3)(27-3) 

=  39-7  kg/cm2  (565  lbs/in2); 

^^^^._^(^^       39-7X16.2^3     ^  (,,7,6  1bs/in2) 

X  10.8 

OhiX—h')  39.7X7.8  .         ,  cy  „        ,.  9^ 

ae'=n-^  ^  =  i5X^^  '     '    =431  kg/cm2  (6130  lbs/m2). 

X  10.0 


SIMPLE  BENDING 


89 


If  now  the  upper  reinforcement  FJ  is  combined  with  the  lower  so  that  only  a 
singly  reinforced  slab  is  secured,  with  7^6  =  28.5+9.5=38  cm^  (5.89  in^)  (see  Fig. 
82),  there  is  obtained 


^^i5X38r_^^/- 2X100X^1, 

100  L      ^  ^  15x38  J 


12.74  cm.  (5.02  ins.) 


so  that 


2ilf  2X600,000  9  /   o     1U    /•  2\ 

(76=  ^=  =4i-S  kg/cm-^  (587  Ibs/m^), 

.    L    x\    100X12.75X22.75  ^  ^  '  ' 

0  X  \  11  —  ' 


Fe 


M  600,000  .         ,       ,        9    /    00      11.     /•  9\ 


By  comparing  the  two  examples,  it  is  seen  that  the  unit  compressive  stress 
is  almost  as  low  w^hen  the  tension  reinforcement  is  increased  by  F/.*  From 
the  standpoint  of  safety  alone,  the  author  prefers  the  proceeding  of  the  last 
example  in  many  cases  instead  of  employing  a  compression  reinforcement, 
because  the  reduction  of  the  steel  stress  ae  means  corresponding  increase  in 
safety,  since  experiment  shows  that  the  compressive  strength  of  concrete  in 
bending  increases  with  the  percentage  of  tension  reinforcement. 


*  As  when  designed  otherwise 


. — Trans. 


CHAPTER  VII 


THEORY  OF  REINFORCED  CONCRETE 

ACTUAL  ULTIMATE  BENDING  TESTS  OF  REINFORCED 
CONCRETE  SLABS  IN  THEIR  RELATION  TO  THEORY 

Tests  of  reinforced  concrete  slabs  have  been  copiously  discussed  in  the  tech- 
nical magazines  *  and  have  been  subjected  to  thorough  theoretical  analysis  by 
Ostenfeld,  v.  Emperger  and  others,  somewhat  along  the  lines  already  indicated. 

It  has  been  found  that  the  compressive  strength  of  concrete  developed,  in 
tests,  increases  with  extra  reinforcement,  because  a  decrease  in  the  ratio  n  seems 
to  take  place  during  the  rupture  stage.  That  is  to  say,  the  increase  in  compres- 
sive strength  was  such  as  might  occur  should  the  steel  stress  exceed  the  elastic 
limit.  In  other  words,  calculations  with  w  =  i5,  in  cases  of  small  percentages  of 
steel  where  it  is  fully  stressed,  do  not  give  correct  results,  so  that  a  lower 
value  for  n  must  be  adopted,  which  will  produce  a  correspondingly  higher  value 
for  Oh. 

When  an  effective  depth  of  \d  is  assumed,  it  is  easy  to  find  the  relation  between 
Fe  and  d  which  will  lead  to  a  minimum  cost,  but  this  condition  is  unattainable 
with  usual  costs  of  materials,  because  with  it  the  safe  compressive  strength  of  the 
concrete  is  exceeded. 

From  a  commercial  point  of  view,  therefore,  the  safe  compressive  strength  in 
bending  is  of  great  importance. 

In  No.  II,  1903,  p.  94  of  Beton  und  Eisen,  v.  Emperger  called  attention  to 
the  fact  that  the  strength  in  compression  of  mass  concrete  derived  from  direct 
pressure  tests  of  plain  concrete  cubes,  should  not  be  used  to  determine  the  safe 
compressive  strength  of  reinforced  concrete  in  bending;  but  rather,  the  actual 
computed  compressive  stresses  derived  from  ultimate  bending  tests,  deduced  in 
the  same  manner  as  those  used  to  determine  theoretical  dimensions.  This  method 
possesses  the  advantage  of  almost  entirely  eliminating  the  effects  of  arbitrary 
inaccurate  assumptions  which  enter  most  methods  of  calculation.  It  can  also 
be  employed  with  any  other  method  of  computation. 

Wayss  and  Freytag  conducted  some  experiments  in  accordance  with  v. 
Emperger's  ideas.  The  concrete  was  mixed  in  the  proportions  of  1:4,  the 
same  as  the  tests  already  described,  and  when  13  months  old  the  specimens  were 

*  G.  A.  Wayss,  "Das  System  Monier,"  1887;  Sanders,  "Beton  und  Eiscn,"  No.  IV,  1902. 
Ostenfeld,  Christophe,  "Beton  und  Eisen,"  No.  V,  1902;  Johannsen-Moskau,  "Beton  und 
Eisen,"  No.  I,  1904. 

90 


TESTS  OF  SLABS 


91 


tested  at  the  Testing  Laboratory  of  the  Royal  Technical  High  School  in  Stutt- 
gart. The  sections  of  three  slab-like  j)ieces,  which  averaged  about  lo  by  31  cm. 
(3.9  by  12.2  in.),  (Fig.  84),  were  2.20  meters  (86.8  in.)  long  and  the  reinforce- 
ment consisted  of  five  round  bars  10  mm.  (f  in.)  in  diameter.  The  other  three 
had  sections  of  about  10  by  25  cm.  (3.9  by  9.8  in.),  (Fig.  85),  and  were  rein- 
forced with  10  round  l)ars  of  10  mm.  (f  in.)  diameter. 

As  is  shown  in  Fig.  83,  a  part  of  the  reinforcement  was  l)cnt  diagonally  upward 
near  the  ends,  to  prevent  a  premature  failure  from  shear.  In  the  test,  the  speci- 
mens were  sup])orted  so  as  to  have  a  clear  span  of  2  meters  (78.7  ins.)  and  the 


P        XQQQ 


"t-  -3/^  ->  <-  Z#S-----¥ 


Fig.  84.  Fig.  85. 


load  was  applied  at  two  symmetrically  located  points  0.50  meter  (19.7  ins.)  apart, 
in  a  continuous  operation  until  rupture  was  produced.  Because  of  the  high  per- 
centage of  reinforcement  employed  (from  1.4  to  3.3%  of  the  section),  in  all  speci- 
mens, the  Ijreak  occurred  at  the  upper  surface,  through  over-stressing  the  concrete 
in  compression.  This  failure  occurred  in  the  vicinity  of  one  of  the  loads  and 
between  the  two  points  of  their  application. 

The  appearance  of  the  fracture  is  shown  in  Fig.  86. 

The  stresses  were  calculated  according  to  method  3,  page  80,  with  n  =  i^, 
the  weight  of  the  specimen  being  taken  into  consideration  as  well  as  the  measured 
loads.  In  the  specimen  31  cm.  (12.2  in.)  wude  with  1.4%  of  reinforcement, 
at  the  occurrence  of  the  first  crack  the  average  load  was  P  =  57okg.  (1254  lbs.) 
for  which 


(7e  =  i57o  kg/cm2  (22,330  lbs/in2), 
(76=92.5  kg/cm2  (1315  lbs/in2). 

In  the  case  of  the  slab  25.1  cm.  (9.9 
in.)  wide,  with  3.3 of  reinforcement, 
the  load  averaged  P  =  io8o  kg.  (2376 
lbs.)  and 

(7e  =  i47o  kg/cm^  (20,900  lbs/in^), 
(75  =  158  kg/cm^  (2247  lbs/in^). 


Fig.  86. 


92 


CONCRETE-STEEL  CONSTRUCTION 


Stage  116  was  really  the  one  involved,  since  the  steel  stress  was  still  within 
the  elastic  limit. 

For  the  breaking  load  there  was  obtained  in  the  same  manner  with  P  =  from 
1444  to  2060  kg.  (3178  to  4534  lbs.) 

with  1.4%  reinforcement:  (7^  =  3800  kg/ cm^,  (7^  =  224  kg/ cm^,    ^^  =  4.2  cm. 

(54047  lbs/in^)  (3186  lbs/in2)        (1.65  in.) 


with  3.3%  reinforcement:  (75  =  2750  kg/cm^,    (75  =  296  kg/cm^,    ^^  =  5.7  cm. 

(391 13  lbs/in^)        (4210)  lbs/in^        (2.2  in.) 

From  these  experiments  is  shown  the  amount  of  increase  in  compressive 
strength  of  reinforced  concrete  with  increase  of  reinforcement.  As  already  stated, 
the  cause  is  to  be  sought  in  the  fact  that  with  low  percentages  of  reinforcement, 
or  with  a  steel  stress  above  the  elastic  limit,  calculations  with  ^  =  15  do  not  give 
correct  results. 

According  to  the  ''Leitsatze,"  one-fifth  of  the  observed  ultimate  strength  may 
be  taken  as  a  safe  working  stress.    On  the  basis  of  the  foregoing  tests,  there  results 


with  1.4%  reinforcement,  ab=  --^=45  kg/cm^  (640  lbs/in^), 

(7e=— ^  =  760  kg/cm2  (10,809  lbs/in^); 

with  3.3%  reinforcement,  ab=  -^  =  59  kg/cm^  (853  lbs/in^), 

...2ZL°.55okg/cmM73..1bs/in2). 

In  the  last  case,  however,  it  is  impossible  to  fully  stress  the  steel,  and  the 
stress  decreases,  the  higher  the  percentage  becomes. 

It  can  safely  be  maintained  that  the  correct  values  have  been  selected  in  the 
"  Leitsatze  "  with  cre  =  iooo  kg/cm^  (14,220  lbs/in^)  and  0.75%  of  reinforce- 
ment together  with  (75=40  kg/cm^  (569  lbs/in^). 

It  may  be  advisable  in  certain  cases,  in  the  compressed  lower  edges  of  beams 
of  variable  depth,  for  example,  to  allow  higher  stresses..  In  such  cases,  however, 
the  steel  stresses  must  be  kept  down  (by  using  greater  percentages  of  reinforce- 
ment) . 

In  addition  to  this  series  of  tests,  another  very  similar  series  was  conducted, 
in  which  the  age  of  the  specimens  was  only  two  months.  At  the  same  time  six 
cubes  of  the  same  age,  made  with  the  same  wet  concrete,  were  prepared. 


TESTS  OF  SLABS 


93 


With  w  =  i5,  the  calculated  stresses  at  the  time  of  the  first  tension  cracks  were 
as  follows: 

with  1.4%  of  steel:  (7^  =  1310  kg/cm^,   ^7^  =  77  kg/cm^, 
(18,632  lbs/in2)      (1095  lbs/iii2) 

with  3.3%  of  steel:  cr^  =  1195  kg/cm^,  (76  =  128  kg/cm^, 

(16,996  lbs/in2),  (1821  lbs/in2) 

At  rupture, 

with  1.4%  of  steel:  (7^=3150  kg/cm^,  (76  =  185  kg/cm^, 

(44800  lbs/ in^),  (2631  lbs/ in2), 

with  3.3%  of  steel:  (76  =  1970  kg/cm^,    (76  =  2ii  kg/cm^, 
(28000  lbs/ in^),      (3000  lbs/ in^). 

Owing  to  the  retention  by  the  cast-iron  moulds  of  too  much  moisture,  the 
compressive  strength  of  the  cubes  was  only  139  kg/cm^  (1977  lbs/in^). 


BENDING  TESTS  OF  CONCRETE  BEAMS  WITH  DOUBLE 
REINFORCEMENT. 

Tests  of  concrete  beams  containing  reinforcement  against  both  tension  and 
compression  are  comparatively  rare.  The  existing  material  is  described  and 
analyzed  in  Nos.  Ill  and  IV,  1903,  of  Beton  und  Eisen  by  v.  Emperger.  The 
conclusion  is  reached  that  an  increase  in  the  compressive  strength  can  be  secured 
by  the  introduction  of  steel  into  the  compression  zone  only  when  such  reinforce- 
ment is  well  anchored  by  a  proper  number  of  stirrups,  so  as  to  prevent  buckling 
of  the  compression  rods,  which  might  otherwise  cause  premature  failure. 

Usually  there  can  be  applied  to  the  calculations  of  doubly  reinforced  slabs 
subject  to  bending,  the  same  formulas  as  for  single  reinforcement,  since  in  most 
cases  the  tensile  strength  of  the  reinforcement  will  determine  the  carrying  capacity. 

It  is  recommended  with  regard  to  compression  reinforcement  of  slabs  and 
beams,  that  the  same  precautions  be  employed  as  in  the  case  of  heavily  reinforced 
columns.  This  should  be  done  at  least  until  by  further  tests  the  accuracy  of 
the  ordinary  methods  of  calculation  has  been  demonstrated.  As  was  shown  in 
the  examples,  it  is  much  better  to  increase  the  tension  reinforcement  than  to  add 
steel  to  resist  compression. 

Where  it  becomes  necessary  to  strengthen  the  compression  zone  of  reinforced 
concrete  beams  because  of  restricted  depth  of  member,  it  can  be  effected  with  the 
greatest  certainty  by  the  introduction  of  spirals  placed  side  by  side  throughout 
the  critical  portions.  This  point  will  be  further  discussed  in  connection  with 
the  subject  of  continuous  beams. 


94 


CONCRETE-STEEL  CONSTRUCTION 


Method  of  Calculation  According  to  Ritter 

In  the  1899  volume  of  the  Schweizerische  Bauzeitung,  W.  Ritter  published 
several  methods  of  calculation,  based  on  various  assumptions,  of  which  the  one 
described  in  the  following  paragraphs  has  found  universal  recognition  in  Switzer- 
land. 

For  the  determination  of  the  position  of  the  neutral  axis,  the  concrete  is 
regarded  as  possessing  tensile  strength  and  the  section  of  the  reinforcement  is 
replaced  by  an  w-fold  greater  concrete  area,  Ritter  then  supposes  the  neutral 
axis  to  pass  through  the  centroid  of  the  imaginary  areas.  He  computes  the 
moment  of  inertia  of  the  section  and  then  calculates  the  compressive  stress  in 
the  concrete  according  to  the  usual  formulas. 

With  regard  to  the  necessary  section  of  steel,  the  assumption  is  made  that 
the  concrete  may  crack  in  tension,  but  that  even  then  the  location  of  the  neutral 
axis  is  unchanged  and  it  therefore  follows  that 

M 


h  —  —  ]ae 
3 


With  the  method  of  calculation  recommended  on  page  87,  the  unit  stress  on 
the  concrete  is  somewhat  lower,  especially  with  deficient  reinforcement,  and  the 
steel  stress  correspondingly  slightly  higher  than  in  the  Ritter  method,  because 
the  arm  of  the  couple  between  the  tensile  and  compressive  forces  is  slightly 
smaller.*  For  ordinary  percentages  of  reinforcement,  the  Ritter  method  can  be 
replaced  for  all  practical  purposes  by  the  old  Konen  method,  because  the  neutral 
axis  lies  very  little  below  the  center  of  the  slab.  In  case  a  safe  compressive  stress 
for  the  concrete  is  assumed,  practical  and  serviceable  results  are  obtainable. 

As  an  example,  the  ultimate  stresses  in  the  previously  described  slabs  have 
been  calculated  according  to  Ritter,  in  order  to  determine  permissible  working 
stresses  to  be  used  with  his  method.  For  the  specimens  with  1.4%  of  reinforce- 
ment (Fig.  87)  and  n  =  2o  (according  to  the  Swiss  "  Normen  ")  the  distance  of 
the  neutral  axis  below  the  center  of  the  slab  is 

d        20X3-93X4  o        /       .  s 

X  =  ^^-^  =  0.8  cm.  (0.31  m.); 

2     31X10  +  20X3.93 
^  =  i(5-83  +  4-23) +20X3-93X3.22  =  3585.6  cm4  (86.1  in4). 

The  breaking  moment  is 

M  =  iii,825  cm. -kg.  (96,856  in. -lbs.), 

so  that  the  compressive  strength  of  the  concrete  amounts  to 

<'i.=^^^^^5^  =  i8o  kg/cm2  (2559  lbs/in^)- 

*  And  the  position  of  the  neutral  axis  somewhat  altered. — Trans. 


TESTS  OF  SLABS 


95 


It  is  224  kg/cm^  (3186  lbs/in-)  according  to  page  92. 

Therefore,  if  according  to  the  German  Leitsiitze 00=  \o  kg/cm-  (569  lbs/in^) 
is  accepted,  then  the  safe  working  stress  according  to  the  Ritter  method  on  the 
basis  of  this  test  will  be 

40X180  .  .  ^ 

—J^-=32  kg/cm-  (455  Ibs/m-). 


According  to  the  Swiss  ''Normen,"  cr6  =  35  kg/cm^  (498  lbs/in^)  is  allowed. 
With  lower  percentages  of  reinforcement,  the  difference  between  the  two  methods 


H 

*  • 

It  .  J/^  ^ 

Fig.  87. 


is  somewhat  greater.  For  instance,  with  0.75%  of  reinforcement,  a  stress  of  40 
kg/cm^  (569  lbs/in^)  calculated  according  to  the  German Leitsatze,"  would  cor- 
respond with  one  of  only  28.5  kg/cm^  (405  lbs/in^)  according  to  the  Swiss. 
''Normen."  It  would  seem,  therefore,  that  their  allowable  working  stress  of  35 
kg/cm^  (498  lbs/in^)  for  concrete  in  bending  is  somewhat  too  high. 

According  to  the  method  and  tables  of  pages  83  to  86,  the  neutral  axis  falls 
slightly  above  the  center  of  the  slab,  whereas  with  the  method  of  calculation  fol- 
lowed in  Switzerland,  it  falls  below  the  center  of  the  slab. 


Position  of  Neutral  Axis 

An  excellent  explanation  concerning  the  position  of  the  neutral  axis  was 
secured  through  some  tests  as  to  the  elasticity  of  reinforced  concrete  conducted 
at  the  Testing  Laboratory  at  Stuttgart. 

The  specimen  shown  in  Fig.  88  was  tested  in  bending,  by  means  of  two  sym- 
metrical loads.    Thus,  a  constant  moment  was  secured  throughout  the  space 


Fig.  88. 


between  the  loads  where  was  located  the  measured  length.  At  each  stage  of  the 
loading,  the  shortening  of  the  upper  concrete  surface  was  measured,  together 
with  the  lengthening  of  the  lower  layer  of  steel.  Because  of  the  constancy  of  the 
moment  and  the  absence  of  cross  stresses  within  the  measured  length,  the  assump- 


96 


CONCRETE-STEEL  CONSTRUCTION 


tion  of  the  conservation  of  plane  sections  during  deformation  was  justified  at  least 
as  long  as  no  cracks  appeared  in  the  tension  concrete. 

Experiments  of  other  testing  laboratories  with  measurements  taken  at  differ- 
ent heights  have  not  shown  this  conservation  of  plane  section.  The  fact  remains, 
however,  that,  could  measurements  be  made  closely  adjacent  to  a  concentrated 
load,  it  would  doubtless  be  found  that  changes  of  length  at  different  heights  were 
not  proportional  to  the  distance  from  the  neutral  axis.  That  is,  because  of  changes 
in  shearing  stress,  neighboring  sections  formerly  plane,  become  curved.*  Meas- 
urements during  stage  lib,  when  isolated  cracks  were  visible,  showed  no  apparent 
irregularity  compared  with  those  of  the  previous  stage.    This  was  probably  due 


Fig.  89. 


to  the  great  measured  length,  80  cm.  (31.5  in.),  so  that  the  effect  of  the  separate 
cracks  was  distributed  throughout  the  whole  length. 

In  Figs.  89  to  91,  the  measured  compression  of  the  concrete  layer  most  distant 
from  the  neutral  axis,  and  the  stretch  of  the  steel  are  plotted  to  a  convenient 
scale — the  figures  employed  indicating  millionths  of  the  length.  The  points  of 
corresponding  strain  are  connected  by  straight  lines  f  corresponding  with  the 
idea  of  the  conservation  of  plane  sections,  so  that  the  location  of  the  neutral  axis 
for  any  corresponding  strains  is  given  by  the  point  of  infersection  of  the  connect- 
ing line  with  the  vertical  representing  the  cross-section. 

*  Compare  v.  Bach  "  Biegeversuche  mit  Eisenbetonbalken,"  Berlin,  1907,  pages  7  and  8. 

t  The  effect  of  the  weight  of  the  specimen  on  the  bending  moment  has  been  taken  into 
account.  Although  but  small  in  itself,  it  was  only  after  this  was  done  that  it  was  possible  to 
secure  a  proper  agreement  with  regard  to  the  stress  distribution  in  the  section. 


TESTS  OF  SLABS 


97 


The  figures  are  the  average  of  three  tests.  It  will  be  seen  that  the  neutral 
axis  is  lower,  the  greater  is  the  amount  of  reinforcement;  but  that  in  all  three 


Fig.  9c. 


varieties  of  specimens  it  moved  upward  with  increasing  load.  Its  initial  posi- 
tion, with  zero  strain,  may  be  determined,  if  in  each  position  of  the  neutral  axis 


98 


CONCRETE-STEEL  CONSTRUCTION 


the  corresponding  moment  is  plotted  upon  a  perpendicular  to  the  cross-section,, 
and  this  moment  curve  is  prolonged  to  an  intersection  with  the  section  line.  The 
curve  thus  obtained  therefore  furnishes  a  picture  of  the  relation  between  the  bend- 
ing moment  and  the  displacement  of  the  neutral  axis.  It  is  shown  in  Figs.  89 
to  91,  as  a  dotted  line.  It  will  be  seen  that  a  Stage  I,  with  a  constant  modulus  of 
elasticity  of  the  concrete  for  tensile  and  compressive  stresses  does  not  exist,  but 
that  with  the  least  loading  an  elevation  of  the  neutral  axis  results. 

With  the  light  reinforcement  of  0.4%  (2  rods  10  mm.  (f  in.))  in  diameter,  the 
initial  position  coincides  almost  exactly  with  the  center  of  the  slab,  whereas  with 
the  heavier  reinforcement  of  1%  (2  rods  16  mm.  (f  in.) )  in  diameter,  it  falls  con- 
siderably below  the  center.  In  all  three  cases  it  coincides  very  closely  with  the 
calculated  position  given  by  the  Swiss  Requirements,  with  n  =  2o.  On  the  other 
hand,  the  highest  (measured)  position  of  the  neutral  axis  corresponds  closely  with 
that  calculated  by  the  German  "  Leitsatze  "  with  ^^  =  15. 

From  the  dotted  line  showing  the  moments  it  can  be  determined  with  cer- 
tainty that  with  increasing  moments,  the  neutral  axis  would  approach  asymp- 
totically a  finite  position  that  would  differ  but  slightly  from  that  obtained  by  cal- 
culation, at  least  as  long  as  Stage  11^,  or  the  elastic  limit  of  the  steel  it  not  exceeded. 
It  can  therefore  be  concluded  that  the  observed  positions  of  the  neutral  axis  in 
sections  with  stress  conditions  intermediate  between  Stages  \\a  and  \\h,  coincide 
with  the  positions  calculated  according  to  the  ''Leitsatze." 

The  exact  location  of  the  neutral  axis  in  the  cross-section  where  cracks  have 
developed  will  probably  never  be  certainly  demonstrated  experimentally.  With 
large  measured  lengths  only  an  average  position  is  obtained. 

Later,  the  calculation  of  the  position  of  the  neutral  axis  for  Stage  \\a 
will  be  considered  on  the  basis  of  the  observed  stress  distribution  in  the  cross- 
section. 

The  tests  under  discussion  afford  a  very  instructive  insight  into  this  stress  dis- 
tribution during  Stage  II. 

Since,  with  the  arrangement  adopted  for  the  experiments,  sections  must  always 
remain  plane  within  the  measured  length,  from  Figs.  89  to  91,  the  deformation 
of  the  concrete  at  any  point  can  be  determined,  and,  with  the  help  of  the  stress- 
strain  curve  made  previously  for  concrete  of  the  same  age  and  composition,  the 
corresponding  stresses  may  be  obtained.  Hence,  for  each  section  there  can  be 
plotted  a  curve  showing  horizontally  the  stress  corresponding  to  each  observed 
deformation  across  the  section  considered  as  axis  of  ordinates  (Figs.  92  to  94) 
and  thus  obtain  for  the  pressure  zone  a  stress  surface,  the  area  of  which  is  equal 
to  the  resultant  compressive  force  D,  which  must  pass  through  its  centroid. 

M 

Since  the  bending  moment  M  is  known,  the  equation  y^~^  gives  the  arm  of  the 

couple  formed  by  D  and  the  tensile  force  Z  which,  with  simple  bending,  must 
be  equal  to  the  compressive  force  U. 

The  tensile  force  Z  is  composed  of  two  components,  viz.,  the  strength  Ze  of 
the  steel  which  can  be  calculated  from  the  measured  stretch  Eg  of  the  steel  and 
its  previously  determined  modulus  of  elasticity  (2,160,000  kg/ cm2  =  30,600,000 
lbs/in^)  and  a  tensile  force  Zb  representing  the  resultant  of  all  tensile  stresses  in 
the  concrete  below  the  neutral  plane.    From  the  known  points  of  application  of 


TESTS  OF  SLABS 


99 


Z  and  Zg,  that  of  Zh  can  be  located.  The  value  of  Z^  must  be  equal  to  the  area 
of  the  tension-stress  surface  of  the  concrete,  and  it  should  traverse  the  centroid 
of  that  area. 

In  Figs.  92  to  94  the  tension-stress  curves  have  been  drawn  as  full  lines  only 
as  far  as  the  observed  stretch  of  the  concrete  corresponds  with  elasticity  tests. 
The  further  presumptive  course  of  the  line  is  shown  dotted. 

When  such  a  course  is  chosen  for  this  line  that: 

1.  The  surface  it  bounds  is  equal  to  Z^, 

2.  Its  centroid  coincides  with  the  computed  position  of  Zt,  and 

3.  The  previously  observed  tensile  strength  of  non-reinforced  concrete  is 

not  materially  exceeded; 

then  it  may  be  concluded  that  the  assumed  course  of  the  line  of  stress  coincides 
with  its  actual  course.  As  may  be  gathered  from  Figs.  92  to  94,  this  coincidence 
is  very  satisfactory  in  view  of  the  variable  composition  of  the  concrete.  It  also 
.applies  to  higher  loads  where  isolated  cracks  have  been  noted. 

Table  XXXII  gives  information  concerning  the  quantities  M,  D,  Z,  Ze,  and  Zb. 
From  the  last  two  columns  of  figures  it  may  be  seen  to  what  extent  the  calculated 
Zh  corresponds  with  the  assumed  value  from  the  tension-stress  surface  of  the 
concrete. 

With  regard  to  the  high  position  of  Zh  in  the  specimens  with  heavy  reinforce- 
ment, it  may  be  noted  that  the  cross-section  of  the  reinforcement  is  to  be  deducted 
from  the  concrete  surface.    All  quantities  are  based  on  a  width  of  i  cm. 


Table  XXXII 


D  from 

Z^  from 

Rein- 
force- 
ment. 

Moment, 
kg-cm. 

the  Stress 
Strain 
Curves, 
kg. 

kg. 

M 
cm. 

Z-Z, 
kg. 

the  Stress 
Strain 
Curves, 
kg. 

First 
Crack 

1992 

96 

51.8=  12 

20.  7 

84 

85 

2826 

134 

87.1=  20 

21  .0 

T13 

117 

i  \ 

3659 

180 

^33-6-=  30 

20.  2 

150 

148 

0 . 

M     (U  1 

4492 

218 

0.105X  2.16X 

206.8=  47 

20.6 

171 

165 

-ods, 
iamet 

5326 

254 

389.8=  88 

20.9 

166 

171 

* 

6159 

323 

649.5=147 

19.2 

176 

180 

6992 

388 

857-8=195 

18. I 

^93 

200 

£  0 

2833 

148 

57-0=  33 

19. I 

115 

98 

4083 

213 

99.8=  58 

19.2 

155 

140 

-0  II 

5333 

269 

157.8=  91 

19.8 

178 

165 

6583 

339 

0. 268X2. 16X  < 

247.4=143 

19.4 

196 

190 

7833 

388 

365. 2=  212 

20.  I 

176 

171 

^  E 
^  S 

9083 

442 

479-5=278 

20.5 

164 

180 

10333 

512 

585-0  =  338 

20.3 

174 

181 

3673 

200 

■   58.7=  65 

18.4 

135 

100 

£^ 

5340 

273 

1 00 . 0  = 1 1 0 

19-5 

163 

137 

7007 

343 

156.0=171 

20.4 

172 

163 

Is,  22 
eter= 

8673 

456 

0.507X  2.16X  , 

224.7=245 

19.0 

211 

191 

* 

10340 

527 

298.0=327 

19.6 

200 

196 

e  1 

12007 

60s 

371.0=407 

19.9 

196 

201 

13675 

685 

442.1  =  485 

20.0 

200 

199 

TESTS  OF  SLABS 


101 


15      30  cm.,  with  varying  pe/ccnlages  of  remforceinent. 


102 


CONCRETE-STEEL  CONSTRUCTION 


The  less  satisfactory  coincidence  in  the  case  of  the  first  loadings  with  heavy 
reinforcement  may  be  explained  as  due  to  initial  stresses  in  the  concrete,  because 
of  shrinkage.  The  measured  tensile  strength  of  i :  4  concrete  in  the  case  of  the 
specimens  used  to  measure  its  elasticity,  Fig.  21,  was  from  8.8  to  10.  i  kg/cm^ 
(125  to  143  lbs/in^).  A  somewhat  greater  tensile  strength  in  bending  in  connec- 
tion with  reinforcement  is  not  surprising,  for  in  that  case  every  eccentric  strain 
is  excluded,  and  a  single  weak  section  can  have  but  a  slight  influence  on  the 
results  of  the  measurements.  A  slight  error  in  D,  with  the  uncertain  elastic  prop- 
erties of  the  concrete,  is  easily  possible,  and  might  produce  a  wide  variation  in  the 
position  and  size  of  Z^. 

In  Figs.  95  to  97,  the  results  of  the  tests  are  shown  graphically  in  the  following 
manner: 

The  moments  (which  were  constant  throughout  the  whole  measured  length) 
are  plotted  as  abscissas.  The  maximum  compressive  stresses  ob,  computed  from 
the  observed  shortening  of  the  edge  of  the  concrete  and  the  known  stress-strain 
curves,  are  shown  as  ordinates  upward.  Downward  ordinates  represent  the  steel 
stress  (7e,  calculated  from  the  measured  stretch  and  the  modulus  of  elasticity 
£e  =  2.i6Xio6  (30,600,000  English  equivalent).  In  this  way  the  curves  shown 
by  heavy  lines  were  obtained.  The  points  at  which  cracks  were  observed  do  not 
correspond  above  and  below,  because  both  curves  are  the  average  of  three  tests 
each,  and  because  the  contractions  and  extensions  could  not  be  measured  simul- 
taneously on  any  specimen.  The  figures  also  show  by  light  lines  the  computed 
stresses  in  the  steel  and  concrete  for  corresponding  moments,  calculated  by  method 
3,  page  80,  with  w  =  15  (corresponding  with  the ''Leitsatze  ").  In  the  same  manner 
the  broken  lines  show  the  results  of  the  Ritter  method  or  according  to  the  Swiss 
''Normen,''  with  w  =  20.  The  diagrams  thus  obtained  are  very  instructive  and 
exemplify  in  a  striking  manner  the  following  deductions: 

T.  First  is  to  be  noted  from  the  sharp  drop  in  the  tension  line  for  light  rein- 
forcement, the  well-known  fact  that  with  slab  reinforcement  below  0.75%  (that 
adopted  in  the  "Leitsatze")  the  safe  working  steel  stress  is  determinative,  while 
with  larger  percentages  of  reinforcement  the  stress  in  the  concrete  is  the  limiting 
factor  in  design. 

2.  The  theoretical  compressive  stress  in  the  concrete,  computed  according  to 
the  "  Leitsatze,"  is  larger  than  the  observed  stress  under  safe  load.  With  heavy 
reinforcement,  the  calculated  value  corresponds  almost  exactly  with  that  found 
by  measurement.  Computations  according  to  the  Swiss  "  Normen  "  give  stresses 
smaller  than  those  actually  observed.  In  Stage  11^,  after  the  occurrence  of  cracks, 
the  Gb  obtained  according  to  the  Leitsatze  "  corresponds  satisfactorily  with 
the  observed  value  (obtained  from  the  longitudinal  measurements). 

3.  The  theoretical  steel  stresses  obtained  by  calculation  are  much  greater 
than  .'ire  actually  observed.  This  holds  good,  of  course,  only  until  the  appear- 
ance of  cracks.  From  that  point,  the  steel  stress  in  the  cracked  cross-sections 
will  be  much  higher  than  in  the  other  parts  and  will  attain  the  values  established 
by  calculation. 

4.  The  curve  of  tensile  stress  takes  the  same  course  as  is  shown  in  the  Con- 
sidere  experiment,  Fig.  50,  page  51.  Table  XXXII,  on  page  99,  shows  in  'figures 
the  same  thing  in  regard  to  the  distribution  of  tensile  stress  Z  between  the  forces 


104  CONCRETE-STEEL  CONSTRUCTION 

Ze  and  Zft.  While  Z  and  Zg  increase  with  increase  of  moment,  Zh^  except  for 
slight  variation,  remains  practically  constant  after  once  attaining  its  maximum 
value.  As  claimed  by  Considere,  therefore,  a  proportional  distribution  of  tensile 
stress  between  steel  and  concrete  must  be  admitted,  but  with  this  difference  from 
Considere's  claim,  that  in  the  tests  here  described,  thanks  to  the  great  care  exercised, 
the  tension  cracks  in  the  concrete  were  discovered  much  earlier.  In  spite  of  their 
existence,  however,  the  distribution  of  stress  remains  the  same,  and  the  tensile 
stress  Zh  suffers  no  material  decrease.  How  can  this  phenomenon  be  explained, 
if  the  ductility  of  concrete  assumed  by  Considere  fails  us? 

According  to  the  records  of  the  tests,  cracks  first  appeared  at  the  pins  A ;  next, 
within  the  measured  length  (the  cracks  w);  and  finally  the  crack  m.  As  the 
lateral  forces  within  the  measured  length  are  nil,  there  occur  during  Stages  I  and 
Ha  within  this  part  no  sHding  stresses.  As  soon,  however,  as  Stage  \\h  is  entered, 
and  a  crack  occurs  in  a  cross-section,  the  reinforcement  is  subjected  at  that  point 
to  more  severe  stresses,  and  in  the  adjoining  sections  the  adhesion  or  rather  resist- 
ance to  sliding  must  assume  its  full  importance  in  the  adjustment  of  stresses 
between  the  concrete  and  steel.  If  africtional  resistance  of  33  kg/cm^  (469  lbs/in^) 
is  assumed,  there  is  obtained  for  the  specimen  with  2  rods  16  mm.  (|  in.)  in  diam- 
eter a  length  of 

15Z6  15X180  f      '  \ 

^  —  -=8.1  cm.  (3.2  m.), 


2X3-14X1.6X33  207 


which  is  necessary  to  restore  in  the  concrete  the  stress  to  which  it  was  originally 
subjected.  Because  of  friction  against  the  reinforcement,  and  of  the  tensile 
strength  which  still  exists  in  the  pieces  lying  between  cracks,  even  cracked  con- 
crete decreases  to  some  extent  the  stretch  of  the  reinforcement.*  Through  these 
causes  is  obtained  an  almost  constant  value  of  Z^,  even  after  the  occurrence  of 
cracks,  as  would  be  obtained  in  conjunction  with  the  phenomenon  of  ductility  of 
concrete,  which,  however,  in  reahty  does  not  exist. 

It  cannot  be  asserted  positively  that  Considere,  in  his  tests,  overlooked  the 
cracks,  but  on  the  other  hand  it  should  be  observed  that  from  the  specimens  of 
the  tests  here  described,  pieces  of  concrete  20  to  40  cm.  (8  to  16  ins.)  in  length 
between  cracks  could  have  been  removed  entirely,  and  they  would  have  displayed 
their  full  tensile  strength.  The  cracks  were  at  first  visible  only  beneath  the  rein- 
forcement, so  that  it  does  not  appear  impossible  that  the  higher  concrete  layers 
might  yet  resist  tensile  stress. 

*  By  employment  of  stretch  measurements  with  small  units  of  measure,  even  the  relative 
displacement  of  the  concrete  with  regard  to  the  steel  can  be  noted.  See  Christophe,  Beton 
und  Eisen,  No.  V,  1902,  p.  14.  On  the  other  hand,  the  use  of  too  small  units  is  the  cause 
of  many  diverse  results  in  otherwise  scientific  experiments. 


TESTS  OF  SLABS 


105 


Safety  of  the  Concrete  against  Tension  Cracks 

5.  Especially  with  light  reinforcement,  the  tensile  stress  taken  up  by  the  con- 
crete relieves  the  steel  to  such  an  extent  that  its  stretch  remains  considerably 
below  the  calculated  figures.  With  more  liberal  reinforcement,  this  is  not  the 
case,  but  here  the  limit  of  compressive  stress  in  the  concrete,  warrants  no  further 
increase  in  the  size  of  the  reinforcement.  Consequently,  when  designing  accord- 
ing to  the  ''Leitsatze,"  i.e.,  according  to  the  conditions  in  Stage  11^,  in  all  cases 
is  obtained  a  factor  of  safety  against  cracking  in  rectangular  slabs  which  amounts  to 

2.12  with  0.4%  of  reinforcement; 
1.50  with  1.0%  of  reinforcement; 
1.64  with  1.9%  of  reinforcement. 

Similar  results  are  afforded  by  the  experiments  described  on  pages  92  and  93,  in 
which  the  computed  unit  stresses  at  the  appearance  of  the  first  crack,  as  com- 
pared with  c»e  =  iooo  and  ^7^=40  kg/cm^  (14,220  and  569  lbs/in^),  give  the  follow- 
ing factors  of  safety  against  cracking  of  the  concrete: 

2.3  with  13  months  old  specimens  with  1.4%  of  reinforcement; 
3.9  with  13  months  old  specimens  with  3.3%  of  reinforcement; 
1.9  with  2  months  old  specimens  with  1.4%  of  reinforcement; 
3.2  with   2  months  old  specimens  with  3.3%  of  reinforcement. 

In  this  connection  is  to  be  noted  other  valuable  material  by  Bach  in  the 
Zeitschrift  des  Vereins  Deutscher  Ingenieure,  1907.  With  regard  to  rectangular 
sections  with  such  reinforcement  as  is  usually  employed  in  practice,  it  is  shown 
that,  with  the  approved  method  of  calculation  which  ignores  tension  in  concrete, 
a  factor  of  safety  is  obtained  of  1.2  to  1.4  against  the  first,  extremely  fine,  almost 
imperceptible  tension  cracks.  The  heavily  reinforced  beams,  however,  {i  and  k 
of  the  quoted  list)  showed  the  first  tension  crack  at  a  computed  steel  stress  of  765 
kg/cm^  (10,881  lbs/in^)  for  the  1.4%  of  reinforcement,  with  a  corresponding 
concrete  compressive  stress  of  45.2  kg/cm^  (643  lbs/in^).  In  this  case  the  com- 
puted stress  was  i.i  times  the  assumed  safe  one.  In  these  cases  the  cracks  were 
so  fine  that  they  could  not  be  observed  with  the  usual  whitened  concrete  surface. 
A  certain  amount  of  practice  was  necessary  to  see  them,  thereby  showing  clearly 
that  in  the  earhest  experiments  of  this  kind  on  similar  specimens,  much  higher 
stresses  actually  existed  when  the  cracks  were  first  discovered. 

It  is  thus  found  from  these  experiments  that  the  customary  methods  of  calcu- 
lation according  to  the  ^'Leitsatze  "  or  the  Prussian  ''Regulations,"  provide  an  aver- 
age factor  of  safety  against  the  appearance  of  the  first  tension  crack  of  1.2  to  1.5. 
Of  course  this  applies  primarily  to  rectangular  sections.  The  application  to  T- 
beams  will  be  considered  later. 


106 


CONCRETE-STEEL  CONSTRUCTION 


The  new  Prussian  "Regulations"  of  May  24,  1907,  in  Sec.  15,  Par.  3,  et  seq., 
require  that  all  buildings  which  are  exposed  to  the  weather,  humidity,  smoke, 
gases,  and  similar  harmful  influences,  besides  being  designed  according  to  Stage 
11^,  shall  also  have  the  added  condition  imposed  that  no  cracks  shall  appear  in 
the  concrete  because  of  tensile  stresses.  The  allowable  tensile  stress  on  concrete 
must  also  be  restricted  to  |  of  that  obtained  by  tension  experiments,  or  to 
of  the  bending  strength,  if  the  tensile  strength  is  exceeded  by  it.  The  prescribed 
method  of  calculation  is  identical  with  that  of  Ritter,  already  explained — that  is, 
the  moduli  of  elasticity  in  tension  and  compression  are  considered  equal  and  con- 
stant, and  the  steel  may  be  replaced  by  a  concrete  area  n  times  larger.  After 
computation  of  the  location  of  the  neutral  axis,  as  the  centroidal  axis  of  this 
modified  section,  the  stresses  can  be  determined  by  the  well-known  equation 


vM 

7 


The  value  1 5  is  selected  for  n. 

There  follow  some  examples  of  the  Stuttgart  experiments  tested  by  this  new 
and  compHcated  method  of  design.* 

The  distance  x  of  the  centroid  of  the  section  shown  in  Fig.  98  from  the  middle  is 


15X2.36X13-5 


^0.75  cm.  (0.295  in-): 


20X30  +  15X2.36 
7  =  -JX2o(i5. 75^  +  14. 25^) +  15X2.36X12.752  =  51,092  cm^  (1226  in^). 


\^ 
1 

1 

1 

1 
1 
1 

r 

A 

1 

 \  

 •  7F  ■  - 

 t  

1 

1 
1 
1 
1 
1 

_  JO 
 »■  - 

1 

K  >\ 

Fig.  98. 


K  -ZO-  ----A 

Fig.  99. 


so  that  the  tensile  stress  on  the  concrete  at  the  appearance  of  the  first  crack  at  a 
moment  of  ilf  =  98,348  kg-cm  (85,183  in-lbs)  was 


14.25X98,348 
51,092 


27.4  kg/cm^  (390  lbs/in^), 


As  a  matter  of  fact  the  tensile  strength  of  concrete  is  only  about  13  kg/cm- 
(185  lbs/in^).  The  foregoing  example  is  of  a  beam  with  only  0.43%  of  reinforce- 
ment, while  the  following  is  for  one  with  1.4%  (Fig.  99).    In  it 


x^ 


15X7-81X13 


=  2.1  cm.  (0.827 


20X30  +  15X7. 81 
/  =  iX2o(i7. 13  +  12.93) +  15X7-81X10.92  =  61,558  cm4  (1477  in4). 

*  See  also  "  Postuvanschitz,"  Beton  und  Eisen,  No.  VI,  1907. 


TESTS  OF  SLABS 


107 


The  bending  moment  at  the  appearance  of  the  first  crack  was  ^1/=  141,010 
kg-cm  (122.134  in-lbs),  so  that  the  computed  tensile  stress  on  the  concrete 
was  approximately 

12.9X141,010  ^  11    /•  2\ 

^2  =  =29.5  kg/cm-  (420  lbs/m2). 


According  to  the  Regulations,"  a  safety  factor  of  ij  against  tensile  cracks 
is  intended,  but  sight  has  been  lost  of  the  fact  that  in  plain  concrete  beams  of 
rectangular  section,  because  of  the  variable  value  of  E,  the  tensile  strength  in 
bending  is  practically  twice  that  found  in  direct  tension  tests.  It  seems  natural, 
and  is  proved  by  these  experiments,  that  the  introduction  of  steel  on  the  tension 
side  makes  very  little  change  in  this  condition.  It  is  thus  evident  that  the  Prus- 
sian Ministerial  ''Regulations"  of  1907  actually  provide  a  three-fold  factor  of  safety 
against  the  appearance  of  the  first  crack,  and  in  consequence  the  execution  of 
reinforced  concrete  work  is  needlessly  costly  and  difficult. 

Of  som.ewhat  more  practical  value  is  Labes'  '"  Vorliiufigen  Bestimmungen  fur 
das  Entwerfen  und  die  Ausfilhung  von  Ingenieurbauten  im  Bezirke  der  Eisen- 
bahndirektion  Berlin  "  (No.  52  of  the  Zentralblatter  der  Bauverwaltung,  1906). 

6i/  . 

In  it  the  bending  strength  0=——  is  taken  as  the  tensile  strength  of  the  con- 

bh^ 

Crete  and  a  factor  of  safety  of  2.5  to  1.3  required.  The  last  value  applies  to  side- 
walks and  light  foot-bridges,  mangers,  water-tanks,  and  structures  subject  to 
slight  vibration.  For  n,  a  value  of  10  is  taken,  since  it  produces  lower  stresses  in 
Stages  I  and  I  la  (strictly,  the  steel  section  should  be  multiplied  by  n  —  i,  because 
of  the  space  displaced  by  it  in  the  concrete). 

The  value  ^^  =  15,  which  is  given  in  the  "Leitsatze"  for  computations  accord- 
ing to  Stage  \lh,  would  not  here  apply,  in  view  of  the  results  of  elasticity  experi- 
ments. It  is  to  be  noted,  however,  that  this  method  of  calculation  does  not 
consider  the  existing  stress  under  the  maximum  allowable  load,  but  rather  a  con- 
dition of  necessary  safety  based  on  stresses  developed  by  much  higher  loads.  It 
is  clear  that  the  value  of  n  should  be  adapted  to  this  later  condition.  For  slabs, 
that  is,  rectangular  sections,  the  factor  of  safety  against  tension  cracks  provided 
by  the  above-mentioned  discussion  is  clearly  superfluous.  The  increasd  safety  is 
secured  through  more  concrete,  which,  however,  at  the  same  time  is  favorable  to 
vibration.  Furthermore,  the  distribution  of  the  reinforcement  tends  to  prevent 
the  appearance  of  the  first  fine  cracks. 


T=BEAMS 


In  T-beams,  subject  to  positive  bending  moments,  the  slab  is  ahvays  made  of 
a  certain  width,  so  as  to  act  statically  with  the  stem,  with  which  it  forms  a  T- 
shaped  section.  If,  however,  the  bending  moment  is  negative,  as  will  be  the  case 
with  beams  anchored  at  the  ends,  or  with  those  passing  over  a  central  support, 
and  again  ignoring  the  tensile  strength  of  the  concrete,  the  calculation  should 
be  made  just  as  if  no  slab  existed.    That  is,  one  should  proceed  in  exctly  the 


108 


CONCRETE-STEEL  CONSTRUCTION 


same  manner  as  indicated  above  for  a  rectangular  cross-section,  but  with  the 
difference  that  the  zone  of  tension  is  found  with  its  reinforcement  in  the  upper 
part,  and  the  compression  zone  in  the  lower  portion  of  the  cross-section,  and 
covering  a  width  equal  only  to  that  of  the  stem  (Fig.  loo). 


jy:  

-yi  JiT 

H  i 

/  1 
/  T 
 j- 

1 

Fig.  ioo. — Distribution  of  stress  with  negative  bending  moments. 


On  the  supposition  that  the  reinforcement  in  the  stem  is  uniformly  distributed 
with  regard  to  the  effective  slab  breadth  b,  calculations  for  positive  bending 
moments  can  be  made  as  for  a  corresponding  rectangular  section,  if  the  neutral 


Fio.  loi. — Distribution  of  stress  with  positive  bending  moments. 


axis  falls  within  the  slab  or  coincides  with  its  lower  edge.  In  the  latter  case,  with 
the  nomenclature  of  Fig.  loi, 


D  =  Z 


from  which 


In  reality,  the  neutral  axis  always  falls  in  the  vicinity  of  the  lower  edge  of  the 
slab.    Whenever  it  falls  somewhat   below  that  point,   as  in   Fig.    102,  the 


TESTS  OF  SLABS 


109 


shaded  portion  of  the  stem  there  shown  (in  which  insignificant  compressive 
stresses  act),  can  simply  be  ignored.  Consequently,  the  centroid  of  compression 
will  be  only  slightly  shifted  from  one  condition  to  the  other. 

If  it  is  considered  that  the  lowest  possible  position  of  this  centroid  can  be  the 
mid-point  of  the  slab  section,  the  maximum  usual  value  of  Z  will  be  given  by 
the  formula 


It  is  thus  seen  that,  because  of  the  small  possible  variation  in  the  location  of 
the  centers  of  tension  and  compression  in  T-beams,  it  is  possible  to  ascertain  the 
tensile  stress  in  the  reinforcement  with  sufficient  accuracy  for  all  practical  pur- 
poses without  recourse  to  special  theoretical  formulas. 





mmm 


•  o  • 


i  ^ 

Fie.  102. — Distribution  of  stress  with  positive  bending  moments  when  x>d. 


The  stress  in  the  concrete  at  the  upper  edge  of  the  slab  does  not  vary  within 
such  small  limits  as  does  the  arm  of  the  couple  of  Z  and  D.  However,  for  cases 
in  which  the  neutral  axis  does  not  fall  within  the  slab,  there  may  be  used  the 
maximum  value 

2Z 
bd  ' 


ab 


or  one  may  proceed  according  to  the  following  more  exact  method. 

The  neutral  axis  is  supposed  to  lie  within  the  stem,  and  at  a  distance  x  below 
the  upper  layer  of  the  slab,  h  is  the  distance  of  the  reinforcement  from  the  same 
layer  and  its  area  is  represented  by  Fe.  The  small  compressive  stresses  in  the 
shaded  area  of  the  stem  are  simply  neglected.  Then,  on  the  supposition  of  a 
constant  modulus  of  elasticity  Eh  of  the  compressed  concrete,  there  is  found,  as 
for  rectangular  sections  (Fig.  102), 

-:^=-:(/,-^), 

from  which  with 

Eb 

there  follows 

_nab(Ji—x) 


110 


CONCRETE-STEEL  CONSTRUCTION 


and  further 


_        bx  ab(x—d)(x—d) 

Oet  e  =  Oh  0   . 


2X2 


Substituting  herein  the  value  of  a^,  gives 

nob{h—x)  bx  (7b(x—d)^ 

■  r  e  =  (7b  ■  J 

X  2  2X 

from  which 

2nhFe+bd^ 


x  = 


2{nFe  +  bd) 


The  distance  of  the  center  of  compression  or  of  the  centroid  of  its  trapezoid 
from  the  neutral  plane,  computed  by  the  equation  of  moments,  is 

d  d^ 
y=x  h 


2       6{2X  —  d) 

In  this  equation  is  clearly  to  be  recognized  for  x=d  the  value 

d     d  d     2  , 

y=x  \-—  =x  =—d, 

26  33 

and  for  greater  values  of  x 

d 

y=x  . 

2 

If  the  center  of  compression  is  known,  the  compressive  stress 

fi—x+y 

as  well  as  the  stress  a^,  and 

OeX       Oe  (2nhFe  +  bd^) 


(76  = 


n{h—x)    n  bd{2h—d) 


can  be  computed. 

The  position  of  the  neutral  axis  may  also  be  obtained  from  the  condition  that 
it  must  pass  through  the  centroid  of  the  modified  section  consisting  of  the  slab 
and  the  w-times  increased  area  of  the  reinforcement.  The  value  of  x  may  be 
immediately  derived  from  the  moment  equation  of  this  area  about  the  upper  edge 
of  the  slab.  From  x,  the  computation  of  y  is  easily  made,  and  then  the  well- 
vM 

known  equation  0=  can  be  employed.    In  that  case 

J 

xM 
(h-x)M 


TESTS  OF  SLABS 


111 


Example  1. — A  reinforced  concrete  beam  28  by  50  cm.  (11  by  19.7  ins.)  stem 
section,  with  a  reinforcement  of  5  round  rods  28  mm.  (ij  ins.  approx.)  in  diam- 
eter, and  a  slab  10  cm.  (3.9  ins.)  thick,  with  an  effective  width  of  250  cm.  (98.4 
ins.)  has  a  positive  bending  moment  of  1,430,000  kg-cm  (1,236,000  in-lbs). 

6  =  250  cm.  (98.4  ins.),    (/  =  iocm.  (3.9  ins.),    /z  =  57cm.  (22.4  ins.), 
F,  =  3o.8  cm2  (4.77  in2),    ^  =  15. 

The  position  of  the  neutral  plane  is  calculated  to  be 


Then 


2X15X57X30-84-250X102 

x=  r — — — 7^  — ^  =  1^.1  cm.  (5.26  m.). 

2(15X30.8  +  250X10)  -  ^ 

10  100  ,     „  .  . 

y  =  '3.i-Y  +  6(,xi3.i-zo)=9-'        (3-58  m.); 

D  =  Z= — ^>43o>ooo — ^^i^Q^^  27,000  kg.  (59,000  lbs.); 
57-13. 1+9.1 

^,  =  '-22^  =  878  kg/cm2  (12,488  lbs/in2); 
30.8 

ob=  878X13-1      ly^g  kg/cm2  (249  lbs/in2). 
15(57-13-1) 

If  the  neutral  plane  had  been  assumed  to  coincide  with  the  lower  edge  of  the  slab,, 
there  would  have  resulted 

^_^_i^43o^_^ 
57-3-3 

(7e  =  864  kg/ cm2(i 2,289  lbs/in2); 

2X26,600  ^  H      /•  9X 

or^v.^  =^^-3  kg/ cm-  (303  lbs/m2). 

250  A  iO 

Example  2. — The  same  beam  is  to  have  double  the  reinforcement  and  be  sub- 
jected to  double  the  moment.    The  slab,  however,  is  to  be  10  cm.  (3.9  ins.)  thick, 
then  equals  61.6  cm2  (0.965  in2),  and  there  results 

2X15X57X61. 6  +  250X102  . 

x  =  ,  , — ;  '  ^ —  =  10.0  cm.  (7.5  m.); 

2X(i5X6i.6  +  25oXio)        ^  ^'  ^  ^' 

100  .        ,       '  s 

y  =  i9.o-5  +  ^^^^^^  ^_^^^  =  i4.6  cm.  (5.75  m.); 

2,860,000  o      n  N 

^^^^57-19.0  +  19.6^54-370  kg.  (119,800  lbs.); 

^.  =  ^^  =  883  kg/cm2  (12,559  lbs/in2); 

^^^_883>09:0  ^       kg/cm2  (418  lbs/in2). 
15(57-19.0) 


112 


CONCRETE-STEEL  CONSTRUCTION 


Example  3. — The  same  beam  as  in  Example  i  is  supposed  to  be  made  of 
concrete  possessing  a  higher  modulus  of  elasticity,  so  that  w  =  io.  Then  there 
follows 

2X10X57X30-8  +  250X102  . 

x=  ,  — r  — ;  =  10.7  cm.  (4.2  m.); 

2(10X30.8  +  250X10)  '  ^' 

^^^"7-T  +  6(2Xio.7-io)=7.2  cm.  (2.83  m.); 

Z=D^-^^^^^^^^^26,.oo\,g.  (58,700  lbs.); 
57-10.7  +  7.2 

(7e  =  ^^^  =  867  kg/cm2  (12,331  lbs/in^); 
30.5 

867X10.7  lU    /•  2N 

oh  =  — ~  ^  =  10.5  kg/cm^  (277  Ibs/m^). 

10(57  —  10.7)      ^  -3    fc"        V  ' '  ^ 


From  the  three  foregoing  arithmetical  examples,  the  following  conclusions 
may  be  derived:  When,  in  a  given  beam,  a  doubling  of  the  reinforcement  makes 
possible  its  carrying  double  the  l)ending  moment,  the  steel  stress  varies  only  to 
an  insignificant  extent,  while  the  stress  on  the  upper  surface  of  the  slab  (when 
the  thickness  remains  unchanged)  increases,  but  to  a  less  extent  than  the  exterior 
forces. 

In  the  examples  given,  the  increase  is  from  17.5  to  29.4  kg/cm^  in  place  of 
17.5  to  35.0. 

This  retarded  increase  in  edge  stress  has  its  origin  in  the  movement  of  the 
neutral  plane  to  a  greater  depth. 

A  similar  effect  on  its  position,  and  in  consequence  on  the  concrete  stress,  is 
caused  by  a  decrease  in  the  modulus  of  elasticity  Eh  (or  an  increase  in  n)  in  such 
manner  that  a  T-beam  of  poor  material  will  show  a  lower  stress  than  one  with  a 
richer  mixture  and  correspondingly  higher  modulus  of  elasticity  Eh  under  other- 
wise similar  conditions. 

The  same  phenomena  also  occur  in  rectangular  sections,  such  as  simple  slabs. 

The  decrease  in  stress  occurs,  however,  much  more  slowly  with  decrease  of 
than  does  the  diminution  in  the  corresponding  compressive  strength,  so  that  there 
is  no  inducement  to  employ  other  than  a  good  mixture. 

Attention  is  again  called  to  the  fact  that  the  simplified  formulas  for  the  cal- 
culation of  T-beams  are  obtained  by  the  somewhat  improper  neglect  of  the 
insignificant  compressive  stresses  in  the  stem,  and  by  the  acceptance  of  a  con- 
stant modulus  of  elasticity  Eh. 

As  to  the  width  of  slab  5,  the  "Leitsatze"  and  the  ''Regulations"  both  stipulate 
that  it  shall  not  be  greater  than  //3,  that  is,  each  side  no  greater  than  //6.  At 
the  same  time  h  should  evidently  not  be  greater  than  the  beam  spacing.  Investi- 
gations concerning  the  effective  width  of  slab  have  not  been  made,  but  in  this 
connection  a  natural  limit  in  the  calculations  is  set  when  the  shear  in  the  two 
vertical  sections  of  the  slab  equals  that  of  the  stem.  More  will  be  said  with 
regard  to  this  point  in  the  chapter  on  shearing  stresses. 

The  permissible  compressive  stress  in  the  concrete  may  be  assumed  as  large 


TESTS  OF  SLABS 


113 


in  T-beams  as  in  those  of  rectangular  sections.  This  maximum  stress  can  be 
employed  in  very  few  cases,  however,  since  too  shallow  and  excessively  reinforced 
beams  would  be  obtained,  which  above  all  are  uneconomical,  and  a  cheaper, 
better  construction  is  produced  with  deeper  beams  and  with  a  stress  in  the  top 
layer  less  than  40  kg/cm-  (570  lbs/in-).*  In  this  connection,  some  authorities 
claim  it  is  of  considerable  practical  importance  that  the  permissible  concrete 
stress  in  the  slab  be  considered  that  stress  which  is  found  by  including  the  effect 
of  a  possible  tensile  stress  in  the  concrete  at  right  angles  to  the  beams  due  to  the 
continuity  of  the  slab.  The  allowable  stress  should  not  exceed  a  theoretical  value 
Oz  . 

<j=(7b-{ — ,  wherein  m  =  i  of  the  coefficient  of  lateral  dilation. 


It  is  the  opinion  of  the  author,  however,  that  this  condition  cannot  be  applied 
to  reinforced  concrete  as  to  homogeneous  materials,  for  in  concrete  all  the  phe- 
nomena of  longitudinal  and  lateral  dilation  differ  from  those  of  isotropic  materials, 
because  of  differences  in  elasticity  and  in  the  ultimate  strengths  in  tension  and 
compression  of  the  former  material.  It  is  very  important,  however,  that  the 
tensile  stresses  in  the  slab  at  right  angles  to  the  beams,  due  to  the  slab  rods,  be 
fully  cared  for.  Obviously,  somewhat  different  is  the  condition  with  regard  to 
girders.  Then,  the  compressive  stresses  of  the  slab  as  a  floor  and  those  of  the 
flanges  of  the  girder  must  be  added.  As  was  said  in  the  Introduction,  in  con- 
sideration of  these  conditions,  it  is  best  to  adopt  a  narrow  w^idth  of  flange  b  in 
computing  girders,  and  furnish  the  slabs  where  necessary  with  haunches  on  the 
beams.  The  simple  addition  of  the  two  compressive  stresses  is  evidendy  not 
rigorously  correct,  since  the  slab,  when  acting  as  a  floor,  is  compressively  stressed 
only  in  the  upper  part,  while  its  stress  in  the  capacity  of  the  head  of  the  T-girder 
is  variable  throughout  the  whole  zone  of  compression.  The  kind  of  stress  in 
such  a  slab  thus  resembles  that  of  bending  with  axial  thrust.  Since  the  above- 
mentioned  suggestion  is  made  purely  on  constructive  grounds,  it  may  well  happen 
that  the  exact  computation  of  the  combined  stresses  may  sometimes  be  abandoned, 
especially  if  more  insight  is  secured  into  the  elastic  deformation  of  a  rectangular 
slab  resting  on  beams  and  girders.  Because  of  the  presence  of  the  distributing 
rods,  a  slab-like  effect  will  always  exist,  in  consequence  of  which  practically  noth- 
ing but  T-girder  compression  stresses  act  in  the  slabs  close  to  and  parallel  with 
a  girder. 

In  the  experience  of  the  author,  tables  and  formulas  for  the  dimensions  of 
T-beams  are  of  small  necessity. 

For  all  cases  where  the  neutral  axis  falls  within  the  slab,  so  that  x'^d,  the 
tables  and  formulas  for  rectangular  sections  can  be  used.    (See  pp.  83  and  85.) 
The  values  of  x  are  there  given  so  that  it  is  immediately  seen  whether  x  is^d. 

The  rib  spacing,  as  a  rule,  is  determined  by  outside  conditions,  and  the  thick- 
ness of  the  slab  depends  on  the  required  carrying  capacity  between  the  ribs.  For 
ordinary  purposes  of  design,  it  suffices  to  determine  the  necessary  area  of  steel 
Fe  with  the  aid  of  the  formula 


m 


Fe  = 


M 


*  See  Morsch,  Deutsche  Bauzeitung,  1907,  Zementbeilagen,  Nos.  11,  12,  13. 


114 


CONCRETE-STEEL  CONSTRUCTION 


If  an  exact  calculation  of  the  stress  is  then  made,  as  a  rule  is  found  smaller 
than  the  allowable  safe  stress. 

If  x-^d,  the  several  quantities  can  be  computed  from  the  following  formulas: 

_2nhFe-\-bd'\ 
^~  2{nFe  +  hdy 

OeX 


Ob^ 


Oe- 


n{h—x) ' 
M 


Also  Ob  may  be  computed  more  accurately  hy  the  formula 

Oe  (2nJiFe  +  bd^) 


Ob=^ 


n    bd{2h-d)  ' 


which  thus  saves  the  computation  of  the  neutral  axis  and  the  centroid  of  compres- 
sion. These  formulas  apply  only  to  the  case  where  the  neutral  axis  lies  at  the 
lower  surface  of  the  slab.  From  them  can  easily  be  computed,  for  rectangular 
sections  and  various  ratios  of  x\Ji,  corresponding  values  of  oh  and  Oe. 

Since  the  steel  section  can  be  computed  very  easily  and  quite  accurately  from 
the  formula 


M 


Oe 


and  where  there  is  no  question  as  to  the  compressive  unit  stress  oh  in  the  top 
layer  of  the  slab  (since  it  will  surely  be  less  than  the  allowable  one),  the  most 
advantageous  information  for  a  designer  is  a  statement  of  the  relation  between 

M 

the  depth  of  beam  li  and  the  given  moment  — ,  which  will  produce  the  allowable 

stress  Ob.    With  this  in  view,  the  diagrams  of  Fig.  103  were  computed. 

The  useful  depths  //  are  taken  as  abscissas,  and  as  ordinates  the  various 

moments  — ,  from  which  are  determined  curves,  which  represent  various  slab 

b 

depths  for  the  stresses  ^7^  =  1000,  (76  =  40  kg/cm^  (14,223  and  569  lbs/in^).  When 
the  points  shown  by  circles  for  each  thickness  of  slab  are  considered,  a  curve  is 
produced.  The  combination  is  a  single  parabola  starting  from  the  axis,  which 
corresponds  with  the  useful  depth  in  rectangular  sections,  and  which  is  repre- 
sented by  the  formula 


^0.39  \y 


These  diagrams  also  include  the  cases  where  the  axis  lies  along  the  under 
side  of  the  slab.    The  dotted  portions  of  the  line  show  that  there  the  slab  thick- 


TESTS  OF  SLABS 


116 


116 


CONCRETE-STEEL  CONSTRUCTION 


ness  is  too  thin  as  compared  with  the  height  of  beam,  so  that  to  be  of  service  the 
size  given  must  be  increased.    If  a  design  is  selected  in  which  the  useful  depth 
h  is  greater  than  that  shown  on  the  diagram,  then  the  stress  oh  need  not  be  com- 
puted, since  it  is  less  than  40  kg/cm^  (569  lbs/in^)  and  consequently  safe. 
The  computation  of  the  curves  was  made  from  the  following  formulas: 


whence  follows 


2nhFe-hb(P 
2(nFe  +  bd) ' 


bd(2X-d)  M 

2{nh—x)  (7e(h—x+yy 


and  with  (7e  =  iooo,  ob= 40,  x^^h;  so  that  with  the  substitution  of  y,  there 
finally  results 

Exact  Formulas  for  T-Beanis 

For  sake  of  completeness,  formulas  which  include  the  compressive  stresses  in 
the  stem  are  here  included.*    In  Fig.  104  the  location  of  the  neutral  axis  is  com- 


0  •  •  • 



1 

1 
1 

1 
1 
t 
( 

Fig.  104. 

puted  as  the  centroidal  axis  of  the  section  formed  by  the  compressed  concrete 
and  the  w-fold  larger  area  of  the  reinforcement,  which  axis  also  forms  the  lower 
edge  of  the  zone  of  compression.  The  equation  of  the  statical  moment  with 
respect  to  the  upper  edge  is 

/,/72  x-\-d 
x{hd^-{x-d)ho^-n{Fe^-Fe')]=—+hQ{x-d)—~+n{Feh-\-Fe'h'), 


and  the  quadratic  equation  for  the  determination  of  :x;  is 

5ox2  +  2x[d{h-hQ)+n  (Fe + Fe')  ]=d%b-  bo)  +  2n{Feh + F/h') . 
*  See  also  Forster,  Forschritte  der  Ingenieurwissenschaften,  1907,  No.  13. 


TESTS  OF  SLABS 


117 


When  X  has  been  ascertained,  the  moment  of  inertia  /  of  the  modified  section 
can  be  computed  with  regard  to  the  axis  NN,  and  the  stresses  calculated  by  the 
well-known  bending  formulas.  Thus 


and 


J =^[hx^  -ih-  bo)  (d  -xf]  +  nFe'  {x  -  h'f  +  nFe  (h  -  x)^y 
xM 


J  ' 


Ge  =  -jr{h—X). 


An  expression  can  also  be  obtained  for  the  distance  y  of  the  resultant  com- 
pressive stress  above  the  neutral  axis,  but  the  equations  given  above  give  a  sim- 
pler, clearer  solution. 

Example. — A  T-beam  of  the  dimensions  shown  in  Fig.  105  carries  a  bending 
moment 

il/  =  8,o2i,ooo  kg-cm.  (694,732  in-lbs.). 


 ->) 

t 

\ 

0.88 
1 

1 
1 

1  — 

-i- 

•  •  •  •  • 

•  •  •  •  e 

I.  According  to  the  exact  formula.  With 

F/  =  o,    Fe=io,  34  mm.  (ij^  inch  approx.)  rods  =90.79  cm^  (14.07  in^), 
bo  =  3,^  cm.  (15.0  in.),     ^  =  160  cm.  (63.0  in.), 
d  =  20  cm.  (7.9  in.),       h  =  i02  cm.  (40.2  in.), 

the  equation 

box^  +  2x[d(b-bo)  -i-nFe\=d^(b-bo)+2nhFb, 

becomes 

38:x;2  ^  2:x:(2oX  1 22  -f  1 5  X90.79)  =  20^  X 1 22  -f  2  X 1 5  X 102  X90.79, 
38jc2  +  76o4X  =  326,61 7. 

Thus 


-7604  + v'7604^ +  4X38X326,617 
2X38 

=  36.4  cm.  (14.33  in.); 
7  =  ^160X36.4^-122  X  16.43) +  15X90.79X65.62 
=  8,253,418  cm4  (83,153  in4), 


118  CONCRETE-STEEL  CONSTRUCTION 

so  that 

8,021,000X65.6  ^  o  lU    /•  9\ 

^^=^5 — Qor.  .tR     ^956  kg/cm2  (13,598  lbs/m2), 

8,021,000X36.4  .     ,      „  .       ^u  I-  9\ 

<Jb=  =35-4  kg/cm2  (504  lbs/in2). 

o,253,4i^> 

2.  Computation  omitting  compressive  stresses  in  the  stem.  Then 

2nhFe+bd^    2X15X102X90.70  +  160X202  ^  .  v 

2(15X90.79  +  160X20)     ^37-5  cm.  (14.76  m.), 

d  400  „  /       .  V 

y=x  —=^7.5  —  10+--  -  =  28.7  cm.  (11.^  m.), 

^        2    6{2x-d)        ^  6(75-20)        ^        \  6 

so  that 

M  8,021,000  ^  lU    /•  9\ 

<7e=7TT^  r-T  =  7  , — ^^  =  946  kg/cm2  (13,455  lbs/m2), 

Fe{h-x+y)    90.79(102-37.5  +  28.7)  ^'        ^  ^'^^^ 

GeX  946X37.5  A      1     /       2  /         n     /•  2\ 

C76=— ,  r  =  -7  ^^-^-^  =  36.7  kg/cm2  (522  lbs/m2). 

n(h-x)  15(102-37.5) 

3.  Computation  according  to  the  simple  approximate  formulas. 

M  8,021,000  .      ^       I         2   /        A        lU  /• 

=90.79(102 -10)  =96°  kg/cm2  (13,654  lbs/m2), 
Oe  (2nhFe+bd^)    960  (2X15XT02X90.79  +  160X400) 


w    bd{2h—d)       15  160X20(204  —  20) 

=37.2  kg/cm2  (529  lbs/in2). 

Although  in  these  examples,  which  are  solved  for  an  actual  case,  the  neutral 
axis  falls  below  the  under  side  of  the  slab,  and  bo  is  small  compared  with  b,  the 
two  approximate  methods  2  and  3  give  differences  in  the  stresses  ae  and  at  scarcely 
worth  mentioning.  Their  practical  value  is  thus  shown.  The  formulas  under 
number  2,  included  in  the  "  Leitsatze  "  and  the  "Regulations,"  were  first  given 
in  the  original  edition  of  this  book  in  1902. 


CHAPTER  VIII 


THEORY  OF  REINFORCED  CONCRETE 


BENDING  WITH  AXIAL  FORCES 

If  the  resultant  of  the  external  forces  intersects  the  cross-section,  the  normal 
components  can  be  replaced  by  an  axial  force  N  and  a  moment  M.  If  the  mod- 
ulus of  elasticity  of  the  concrete  is  accepted  as  a  constant  for  the  calculations,  and, 
further,  as  often  happens,  a  compressive  force  N  is  involved,  two  cases  are  to  be 
distinguished.  Consideration  will  here  be  given  only  to  rectangular  cross- sections, 
since  for  irregular  sections  the  graphical  treatment  given  later  is  preferable. 

I.  Only  compressive  stresses  are  supposed  to  act  over  the  whole  section.  By 
the  centroid  O  of  Fig.  io6  is  understood  the  centroid  of  the  section  produced 


— 7-  T — : 

i                  """1  " 

i  ^ 

a  i.^ 

i  > 

1 

1 
1 

.  ^ 

I 

\ 

i                         C  ~ 

(5^  \ 

Fig.  106. 

M 


when  to  the  concrete  area  is  added  that  of  the  reinforcement  multiplied  by  n  = 


If  for  I  cm.  width  of  section  fe  represents  the  area  of  steel,  so  that  fe^~^  ^^'^ 
F  ' 

//  =  —-,  the  location  of  the  centroid*  is  given  by  the  formula 


J2 


z 

The  compressive  stresses  produced  by  the  normal  force  AT"  acting  at  the  cen- 
troid are  distributed  uniformly  over  the  entire  concrete  section,  so  that 

A"   

hd^niFe^F^y 

*  Below  the  top  laver. — Trans. 

119 


120 


CONCRETE-STEEL  CONSTRUCTION 


The  moment  M,  with  reference  to  the  centroid  of  the  modified  section,  pro- 
duces on  the  one  side  compressive  stresses  and  on  the  other  side  tensile  ones. 
In  this  case,  however,  the  tensile  stresses,  since  they  represent  only  a  decrease  in 
the  uniformly  distributed  compressive  stress,  are  to  be  calculated  as  for  a  homo- 
geneous section,  in  which  the  area  of  reinforcement  is  to  be  replaced  by  a  concrete 

one  —  times  larger.    It  is  thus  necessary  to  calculate  the  moment  of  inertia  / 
Eh 

in  the  formula 


vM 


from  the  expression 

J  =  -u^  +  -(d- «)3 + n  Feih  -uY+  nF/ (u - h')"^. 
3  3 

Bending  with  axial  compression  is  the  usual  stress  condition  in  the  sections  of 
arches.  In  them  the  reinforcement  is  usually  symmetrically  arranged,  so  that  the 
centroid  of  the  whole  section  coincides  with  the  axis  of  the  arch  and  the  calcu- 
lation assumes  a  fairly  simple  form.    The  area  of  the  modified  section  is  then 


and  the  moment  of  inertia  is 


F  =  bd  +  2  n  Fe, 


J  =  —d^  +  2nFe(--cy. 

12  \2 


If  values  are  assumed  for  F  and  /,  the  same  conditions  exist  in  the  reinforced 
section  as  regards  the  rib,  as  for  a  homogeneous  section. 

2.  The  resultant  is  supposed  to 
have  such  an  eccentricity  that  tensile 
stresses  exist  on  one  side  of  the  section. 

If  these  tensile  stresses  are  in- 
significant, the  calculation  may  be 
made  exactly  as  in  i.  If,  however, 
they  are  appreciable,  a  special 
modulus  of  elasticity  for  tension 
must  be  introduced  into  the  calcu- 
lations. Usually,  in  order  to  obtain 
a  proper  safety  factor,  the  tensile 
strength  of  the  concrete  is  disregarded,  SLb  in  simple  flexure. 

In  Fig.  107,  O  represents  the  centroid  of  the  concrete  section  to  which  the 
moment  M  is  referred,  and  x  is  the  distance  of  the  neutral  axis  from  the  com- 
pression edge  of  the  section.  Then 


Fig.  107. 


N=--bx  +  Fe'Oe'-Feae,  (t) 

2 


M  = 


ohh  X  (d 


---)+Fe'aJe'  +  Fe(Jee  (2) 

23/ 


BENDING  WITH  AXIAL  FORCES 


121 


Further,  because  of  the  conservation  of  plane  sections, 

d  d 
^    e-\  X         e-\  X 

Ee  2  2 

Oe=-prrsb  — -  =  nab  — ,   (3) 

J^l)         oc  oc 

^    e'  \-x  e'  

,     Ee  2  2 

''^=E,''^~^=''''—^— 

These  four  equations  suffice  for  the  determination  of  the  four  unknowns,  x, 
ohy  (7e,  Oq.  If  the  external  forces  and  given  dimensions  are  used  to  calculate  x 
the  following  equation  of  the  third  degree  results,  which  can  best  be  solved  by 
trial. 


Then 

Nx. 


"'^^^  ' — d — ^ — rr^ 


'^+nF/(e'- 


+xj  —nFe{e-\  X 


As  a  rule,  in  arches  and  columns,  the  reinforcement  is  symmetrically  arranged, 
and  there  are  obtained,  from  equations  (i)  to  (4),  with  Fe  =  Fe  and  e'=e,  the 
following  relations: 

A^  =  ^6^  +  F.K-.7.),  (5) 

M  =  o^~i~-''~\-\-eFe{aJ\o^,  (6) 


2    \2  3 
d 

e-\  X 

2 

Ge  =  nOb  -,  (7) 

X 


e  \-x 

2 

Ge^nob  ,  (8) 

X 


while  the  equation  for  the  solution  of  x  takes  the  form 

-x^  (n-  -—\+2X  M  n  4-'   fi^ {M  d  +  iNe^)  =  o, 
6        \    4      2  /  0  0 


122 


CONCRETE-STEEL  CONSTRUCTION 


or 


This  equation  may  be  solved  l^y  any  approximate  method,  or  directly. 
If,  as  is  known,  there  is  assumed  in  the  general  cubic  equation 

+  cix^  -\-hx  ■\-c^o, 

the  new  relation  x  =  z  —  ^a,  there  is  obtained  a  reduced  cubic  of  the  form 

z^-\-  pz  +  q  =  o, 

from  which,  according  to  Cardani's  formula,  may  be  derived 

z=|/-k+v(k)-+(i/')' 


With  the  values  of  equation  (9)  the  reduced  cubic  becomes 

¥1  4nFe  M 

N'^'^  [4     NJ       b  N 


2  M  .  AfiFt 


i2nFee^ 

— y-=°' 

from  which  it  follows  that 

d  M 
x=z  +  ----. 

2  J\' 


z  here  represents  the  distance  of  the  neutral  axis  from  the  point  of  application  of 
the  resultant  normal  force. 

When  X  is  ascertained,  the  stresses  may  be  found  by  inserting  the  value  of  x 
in  equations  (8),  (7),  (6),  and  (5),  and 


Ob 


bx  ,  nFe, 

 1  (2X-  d) 

2  X 


d 

e-\  X 

2 

Ge  =  n  Ob   . 

X 


d 

e  [-X 

Oe—nob  -o 

X 


BENDING  WITH  AXIAL  FORCES 


123 


The  process  is  somewhat  complex,  and  is  not  simplified  when  the  reinforcement 
on  the  compression  side  is  left  out  of  consideration,  so  that  F/=o. 

In  practical  cases,  especially  when  the  amount  of  reinforcement  must  first  be 
determined,  a  l»riefer  method  may  be  followed:  Compute  the  edge  stress  as  for  a 
homxOgeneous  cross-section  without  reinforcement,  as  in  the  case  of  a  rectangle. 


Ob 


X  ,  6M 


(7z 


bd  bd^' 


Then  suppose  all  the  tensile  stress  in  the  section  carried  by  the  reinforcement, 
the  strength  of  which  must  therefore  be 


Further  (Fig.  io8). 


so  that 


and 


Z^-ib(7z{d-x). 


d  6M 
2  bd"- 


d—x  = 


Ozd^b 


12M' 

24M 


is  obtained. 

Furthermore,  approximately, 


Fig.  108. 


Fe' 


When  the  edge  stress  from  the  rib  moment  has  been  obtained,  Z  can  be  cal- 
culated as  the  area  of  the  tension  surface. 

Example. — Assume  a  rectangular  section  in  which  b  =  i  cm.  (0.4  in.)  and  d^ 
90  cm.  (35.4  in.)  for  which  ilf  =  30,000  cm-kg  (25,984  in-lbs),  A^  =  66okg.  (1452 
lbs.),  Fe=  0.37  cm-  (0.057  'm-)=Fe,  e^e'  =  /[o  cm.  (15.7  in.),  ^  =  15. 

According  to  (9)  there  is  obtained 


\  2  660 


•?o,coo  O.S7 
+  :x  X  12  X^-^— X  15 

660        ^  I 


1^X0.^7/^0,000 
■6X^-^(^^^90  +  2X40-  )  -o, 


660 


or 


^3  +  i.364.\;2  +  302  7.3Jt;-  242,} 73.65  =c 
of  which  the  root,  found  by  the  method  described  alcove,  is 


jc  =  46.3  cm.  (18.65  i^-)- 


124 


CONCRETE-STEEL  CONSTRUCTION 


From  this,  according  to  equation  (lo),  is  found 

(Jb^^—  =  28.2  kg/cm^  (401  lbs/in^), 

and  according  to  (7), 

*  c7e-i5X28.2^'^"^^^~^^-3^354  kg/cm2  (5035  lbs/in^), 
40.3 

cT/  =  i5X28.24^^4|+46^_^^g  j,g^^^2  (33^6  lbs/in^). 

The  approximate  method  would  have  given. 

660      30,000X6  lu  /•  ox 

(76=-  \--  =  2Q.6  kg/cm^  (421  Ibs/m^), 

1X90    1X90X90     ^      t,/  / 


and 


so  that 


(72=-- 14.9  kg/cm^  (212  lbs/in^), 

^     62^/3^,2     1X903X14.9'  1       /     QAIK  N 

Z  =  —  =  ^  =  224  kg.  (3186  lbs.), 

24 .  M      24X30,000        ^   &  /> 


2  24 

<7e=^—  =  about  600  kg/cm2  (8534  lbs/in2). 


The  approximate  method  thus  gives  an  almost  identical  result  for  the  com- 
pressive stress  Ob,  but  one  that  is  too  far  at  variance  for  the  steel  stress  cr^. 

For  arch  ribs,  the  rib-moments  are  first  to  be  ascertained  by  customary 
methods,  and  from  these  moments  are  then  to  be  computed  the  axial  force  N 
and  the  moment  M  with  reference  to  the  centroid  of  any  cross-section,* 

w  ~F'^ir 

Mko    N  M 


Ou- 


w    F  ir 

from  which,  through  addition  and  subtraction  of  these  equations, 

(7o  +  (7u  Mku—MkOj^ 

A  =  P  =   Pj 

2  2  IT 

Oo  —  Oujj.     Mko  +  Mku 


so  that  the  stresses  ab  and  ae  can  be  computed  exactly. 

*  W  =  Section  modulus  of  rib  at  point  in  question. 

i*'  =  Modifie(i  area  of  section. 

<Tti  =  Unit  stress  of  extreme  inner  layer  of  rib. 

<To  =  Unit  stress  of  extreme  outer  layer  of  rib. 
lfA;M  =  Moment  from  loading  producing  stress  in  inner  ayer. 
Mfco  =  Moment  from  loading  producing  stress  in  outer  layer. — Trans. 


BENDING  WITH  AXIAL  FORCES 


125 


Fig.  109  is  a  diagram  which  obviates  the  necessity  for  the  solution  of  the 
cubic  equation.    It  is  evident  from  eciuation  (9)  that  with  given  measurements, 

X  is  dependent  only  on  the  ratio  — ,  which  represents  the  eccentricity  of  the 

M 

normal  force.    —  in  (9)  can  therefore  be  expressed  as  a  function  of  x  and  there  is 

N 

obtained 

3     ^  nFeC^ 
-x^+-dx-  +  i2—r~ 
M  2  h__^ 

N       o    12  X  n  Fe     6dn  Fe' 

3^2+  

If  the  area  of  reinforcement  is  expressed  in  parts  of  the  concrete  section,  so 
that 


and  further  if 


then,  with  n  =  i$, 


Fe  =  Fe'=iibdy 
e  =  o.42d, 
M     —  x^-\-^dx^  + 7,1. J  ^/id^ 


(II) 


Nd    ^x-^d  + 1  Sox fid^ — goiid^ ' 

,  M 

With  assumed  percentages  of  reinforcement,  the  ratio        can  be  computed 

for  various  values  of  x  expressed  in  terms  of  d,  such  as  x  =  o.id,  o.2d,  etc.  If. 
in  a  system  of  coordinates,  the  values  of  x-==-o.id,  o.id,  etc.,  are  taken  as  abscissas 

M 

and  as  ordinates  the  values  of         for  assumed  percentages  of  reinforcement, 

curves  are  obtained.  With  their  help,  together  with  the  known  values  of  M  and 
A^,  and  assumed  values  for  d  and  F^=ubd,  the  distance  x  of  the  neutral  axis 
from  the  compression  edge  is  immediately  found  in  terms  of  d.  The  series  of 
curves  shown  in  Fig.  109  is  obtained  for  different  percentages  of  reinforcement 
of  /i  =  o.ooi  to  0.05,  or  0.1%  to  5%.  These  curves  permit  of  ready  interpolation 
of  intermediate  values  of  /i.  The  employment  of  this  table  materially  facilitates 
calculation.    After  x  is  ascertained  the  stresses  are  computed  as  follows: 

N  2Nx 

Ob  = 


bx    Fefi^  bx^  +  2abdn(2X—d)' 

 1  (2X—d) 

2  X 

d 

e-\  X 

2  O.Q2d—X 

Oe=nab  =  I  ^ob  , 

.  X  ^  X 

e  \-x  , 

2  x—o.oQd 
Oe —nob  =  15^^^ 


X  X 


the  last  values  resulting  if  e  is  made  equal  to  0.42^^. 


126 


CONCRETE-STEEL  CONSTRUCTION 


M 

From  the  course  of  the  curves,  it  follows  that  for  certain  small  values  of   , 

Nd 

no  point  of  intersection  with  the  curves  is  possible.  In  other  words,  the  eccen- 
tricity of  the  compressive  force       is  then  so  small  that  the  neutral  axis  falls 

M 

outside  of  the  section,  and  no  tensile  stress  exists  in  it.    The  ratio  of  —  expresses 

the  distance  of  the  resultant  force  from  the  centroid  of  the  cross-section,  and  it 
is  seen  *  that  the  smaller  the  percentage  of  reinforcement  the  nearer  the  value 
of  X  approaches  to  \d  =  o.i6']d,  while  for  heavier  reinforcement  this  value  is  but 
slightly  exceeded.  The  curves  have  vertical  asymptotes  which  correspond  wdth 
the  positions  of  the  neutral  axis  in  simple  flexure,  while  a  common  asymptote  for 
all  the  curves  is  the  straight  line  obtained  by  making  //=o. 

The  framing  of  dimensioning  formulas  is  of  little  value,  because  flexure  with 
axial  compression  occurs  in  most  instances  in  statically  indeterminate  structures, 
such  as  some  arches,  trusses,  etc.,  the  sections  of  the  parts  of  which  must  be 
assumed  so  as  to  calculate  the  external  forces  M  and  A^.  Consequently,  the  only 
operation  is  the  testing  of  the  dimensions  chosen.  Usually  such  statically  inde- 
terminate structures  are  calculated  with  respect  to  the  external  forces  without 
regard  to  the  reinforcement  of  the  members,  after  which  the  necessary  amount 
of  steel  is  computed  by  the  approximate  methods  given.  The  exact  determina- 
tion of  the  stresses  can  follovv^  afterwards. 

In  bending  with  axial  compression  the  allowable  compressive  stresses  of  the 
concrete  and  the  steel  can  be  simultaneously  safely  employed  only  with  a  certain 

M 

eccentricity  of  the  normal  force.    The  relation  between  the  value  of  the 

^  Nd 

eccentricity  of  the  normal  force  and  the  corresponding  percentage  of  reinforce- 
ment [1  may  be  obtained  as  follows: 

For  unit  stresses  ^7^=40  and  (7^  =  1000  kg/ cm^  (569  and  14,223  lbs/in^),  and 
w  =  i5,  according  to  equation  (7) 

8  \  2 

and  with 

Fe=Fe  =  fjhdy    and  e=o,/[2d 

there  follows  from  equations  (5)  and  (6) 

7V=4o&  ^(0.1725-13.478/^), 
M  =  4ob     (0.06641  -}-  i5.34«), 


so  that 


M  0.06641 -|- 1 5.34// 
Nd    0.1725  — 13. 478/{* 

M 

If  the  values  of  —  are  taken  as  ordinates,  and  as  abscissas  the  values  of  ft 


Nd 


From  the  figure. — Trans. 


BENDING  WITH  AXIAL  FORCES 


127 


used  in  this  equation,  the  curve  of  Fig.  no  is  obtained.  For 


0-5 


O.Q2 


o,  and  reinforcement  is  unnecessary, 


as  IS 


M 


naturally  also  the  case  with  values  of  j^^^vS^S- 


However, 


for  the  sake  of  safety,  some  reinforcement  should  generally 
be  used  when  the  point  of  application  of  the  normal  force 
falls  only  o.iis^i  from  the  edge.  And  when  such  eccen- 
tricity exists  and  only  insignificant  reinforcement  is 
necessary,  it  is  advisable  to  ignore  the  theoretical  amount 
and  use  more  steel,  since  its  cost  is  small,  especially  when 
the  eccentricity  may  increase  for  some  unknown 
reason. 

With    /(=i.289o   '-^  vertical  asymptote  to  the 
curve    exists.     That    is,    at  / 


that  value 


00,  or  only 


pure  bending  exists.  In  this 
case  with  e  =  o.^2d,  the  con- 
dition is  one  of  flexure  in  a 
rectangular  section  with  sym- 
metrically placed  double  re- 
inforcement. 

It  is  to  be  noted  that 
from  the  formulas  for  bend- 
ing with   axial  compression, 

all  those  for  pure  flexure  can  be  secured  by  making  N 


yz>-o  o,/  0,2  0,5  0,4^  c>,s  o,e  o,?'  o,<f  0,9  10  /,/  /,2  /,2<f % 

Fig.  1 10. — Diagram  of  the  relation  between  the  eccentricity 
of  the  normal  force  and  the  percentage  of  reinforce- 
ment with  combined  maximum  allowable  stresses,  of 
<7  =1000,  (76=  40  kg/cm^  (14,220  and  569  lbs/in^). 


BENDING  WITH  AXIAL  TENSION 

In  the  employment  of  reinforced  concrete  for  silos,  there  occurs  the  condi- 
tion that  an  axial  tension  N  acts  in  addition  to  a  bending  moment  M.  The  case 
of  bending  with  axial  tension  will  therefore  be  discussed. 

In  Fig.  Ill,  where  a  rectangular  section  is  assumed  of  breadth  6, 


and 

Further 


N  =  FeOe  X  —  FeOe, 

2 


2    \2  3 


d 


128 


CONCRETE-STEEL  CONSTRUCTION 


These  four  equations  correspond  with  the  equations  (i)  to  (4)  for  bending 
with  axial  compression,  except  that  N  is  here  introduced  as  a  negative  quantity. 
Thus  (with  the  supposition  of  symmetrically  arranged  reinforcement,  wherein 
Fg  =  FJ  and  e=e'),  there  may  also  be  obtained  a  formula  for  the  calculation  of 
X  from  equation  (9),  if  N  is  therein  inserted  as  negative.  Then 

Further,  there  is  obtained  for  the  calculation  of  the  series  of  curves  which 
obviate  the  solution  of  the  cubic  equation,  the  relation 

—M_  —x^+^dx^  +  7,i.j^/xd^ 
Nd    2>^^d  + 1  Soxjuid^ — gojud^^ 

corresponding  to  equation  (11),  with  the  nomenclature  there  employed.  The 
curves  (see  Fig.  109)  are  therefore  the  branches  lying  on  the  negative  ordinate 
side  of  the  axis  of  abscissas,  which  correspond  with  those  on  the  positive  side, 


—  { 

1 

1 

e' 
1 

jr.  J 

1 

1 

1 

\) 

— V 

V 

Fig.  III. 


and  complete  each  a  curve  of  the  third  degree.    The  negative  branches  have 
vertical  asymptotes  in  common  with  the  corresponding  positive  branches  and  for 
x=Q  all  curves  for  different  values  of  /i  pass  through  the  point  with  the  ordinate 
,3 1  •  7  S 

 under  which  condition,  with  ^c==o,  the  concrete  is  theoretically  useless. 

Nd     90  ^ 

In  addition,  at  this  point,  all  the  negative  branches  have  a  common  tangent,  the 

inclination  of  which  is 

tana-— ^. 

After  X  is  determined,  all  the  stresses  can  be  ascertained  from  the  following 
formulas: 

—  2Nx 


Oh- 


hx^^2pbdn{2X—d) ' 
o.^2d—x 


X 


x—o.oSd 


Ge  =  1 5^76- 


BENDING  WITH  AXIAL  FORCES 


129 


In  flexure  with  axial  tension  also,  Z  and  can  be  calculated  approximately 
by  the  formulas 


from  which  flexure  with  axial  compression  is  computed. 

M  31 -7  S 

For  values  of  smaller  than        ^  =0.3528,  the  axis  falls  outside  the  section 

A  d  90 

and  the  tensile  force  is  then  to  be  divided  according  to  the  law  of  the  lever, 
if  the  condition  that  the  concrete  is  to  carry  no  tension  is  maintained.  Tal)le 
XXX  gives  a  comparison  between  results  obtained  by  the  exact  and  the  approx- 
imate methods. 


Table  XXX 


Kind. 

h 

d 

71/ 

M 

e 

cm. 

in. 

cm. 

in. 

kg. -cm. 

in. -lbs. 

kg. 

lbs. 

Nd 

cm. 

in. 

cm2 

in2 

Pending 
with 
axial 
com- 
pression 

I 
I 
I 

0.4 
0.4 
0.4 

50 
50 

19. 
19. 
19. 

7 
7 
7 

7500 
I  2500 
15000 

6496 
10826 
12991 

500 
500 
300 

1102 
1 102 
661 

0.30 
0.50 
1 .00 

21 
21 
21 

8.3 
8-3 
8.3 

0.15 

0-15 
0.30 

0.023 
0.023 
0.046 

Bending 
with 
axial 
tension 

I 
I 
I 

0.4 
0.4 
0.4 

16.5 
12.0 
27.0 

6. 

4- 
10. 

5 
7 
6 

1520 

713 
4000 

1316 
617 
3464 

45-6 
25.6 
106.4 

1005 

564 
2346 

2.02 
2.32 
1 .40 

6.92 
5.00 

11-35 

2-7 

2.0 
4-5 

0. 165 
0.103 
0.  290 

0.026 
0.016 
0.045 

Exact. 

Approximate. 

Kind. 

X 

Oh 

Oe 

Gh 

Oe 

cm. 

in. 

kg/cm- 

lbs/in  2 

kg/cm  2 

lbs/in  2 

kg/cm2 

lbs/in  2 

kg/cm  2 

lbs/in  2 

Bending 
with 
axial 
com 

pression 

0.003 
0.003 
0.006 

35-8 
23-3 
19.8 

14. 1 
9-2 

7-8 

25-9 
44-2 
39-5 

368 
629 
562 

1 10 
6^8 
782 

1565 
9217 
11122 

28.0 
40.0 
42.0 

598 
569 
597 

296 
IITO 
1041 

4210 

15787 
14806 

Bendnig 
with 
axial 
tension 

0..0100 
0.0086 
0.0107 

4.45 
3.20 

6.75 

1-75 
1 . 26 
2.66 

22.8 
23.1 
20. 1 

324 
329 
286 

826 
850 
820 

1 1 748 
12090 
IT663 

30-7 
27.6 
29.0 

437 
393 
413 

982 

997 
923 

13967 
14180 

13138 

The  last  three  cases  represent  practical  examples  of  silos  actually  erected. 

The  results  of  the  calculations  can  be  readily  tested  as  to  their  accuracy  by 
introducing  numerical  values  in  equations  (5)  and  (6)  and  noting  whether  the 
equations  are  satisfied. 


130 


CONCRETE-STEEL  CONSTRUCTION 


GRAPHICAL  METHODS  OF  CALCULATION 

All  the  foregoing  discussion  applied  primarily  to  rectangular  or  T-shaped 
sections.  It  is  quite  possible,  however,  that  sections  of  other  shapes  may  occur — 
circular,  annular,  etc.  For  such  cases,  the  formulas  to  be  deduced  would  be  very 
complex  at  best.  The  following  graphical  methods  are  therefore  recommended. 
They  lead  directly  to  the  desired  result  in  a  simple  manner  and  for  any  desired 
rorm  of  section.  In  the  two  methods  given  on  pages  12  and  13,  a  treatise  by 
Autenrieth    of  Stuttgart  will  be  followed. 

{a)  Simple  Bending.— li  no  normal  force  acts  on  a  reinforced  concrete  sec- 
tion, only  a  bending  moment  M  existing;  and  on  the  assumption  that  sections  of 
steel  after  deformation  remain  in  the  corresponding  planes  with  the  compressed 
concrete  sections;  and  from  the  equahty  of  the  tensile  and  compressive  forces, — 
it  follows  that  the  neutral  axis  (which  at  the  same  time  limits  the  compression 
zone  of  the  concrete  section)  is  the  centroidal  axis  of  the  area  consisting  of  the 
compressed  part  of  the  section  and  the  w-fold  increased  steel  section  on  the  ten- 
sion side.    (See  also  page  81.) 

This  surface  is  known  as  the  modified  cross-section.  It  follows,  moreover, 
that  for  this  modified  section,  the  stresses  can  be  calculated  according  to  Navier's 
bending  formula 

vM 

since  the  quantities  are  identical  with  those  of  a  homogeneous  cross-section, 
wherein  the  area  of  the  tensile  steel  is  replaced  by  an  w-fold  greater  concrete 
area.    The  stress  on  the  steel  is  then 

V  M 

(7e=n  a  =  n—j-, 

where  /  is  the  moment  of  inertia  of  the  imaginary  area,  computed  for  the  neutral 
axis  passing  through  the  centroid. 

For  a  symmetrical,  otherwise  unrestricted  cross-section,  the  axis  of  sym- 
metry of  which  falls  in  the  plane  of  the  forces,  two  force  polygons  I  and  //, 
with  equal  polar  distances  H,  are  to  be  drawn,  starting  from  B  and  A,  so  as  to 
make  a  line  polygon  (Fig.  112).  The  loads  which  form  the  line  polygon  BD 
for  the  zone  of  compression  are  to  be  made  up  from  strips  of  the  compressed  area 
taken  perpendicular  to  the  axis  of  symmetry,  and  in  the  polygon  AD  for  the 
zone  of  tension,  of  the  w-fold  increased  area  of  the  reinforcement.  When 
reinforcement  is  found  in  the  zone  of  compression,  as  may  often  occur,  the 
polygon  BD  is  to  include  the  w-fold  increased  steel  area,  beside  the  strip  in  which 

*  "Berechunung  der  Anker,  welche  zur  Befestigung  von  Flatten  an  ebenen  Flachen  dienen." 
Zietschrift  des  Vereins  Deutscher  Ingenieure,  1887.  The  treatise  does  not  relate  directly  to 
reinforced  concrete,  but  the  conditions  are  identical  with  those  here  discussed,  so  that  the  methods 
can  be  appHed,  without  change,  to  reinforced  concrete. 


BENDING  WITH  AXIAL  FORCES 


131 


it  acts.  Now  it  is  known  that  the  moment  of  a  system  of  parallel  forces  with 
respect  to  a  parallel  straight  line  is  equal  to  the  }:)ortion  of  the  straight  line  inter- 


i 


Fig.  112. 


cepted  by  the  two  external  sides  of  a  line  poylgon,  multiplied  by  the  horizontal; 
H,  of  such  line  polygon.    Thus,  referring  to  Fig.  113 


and  applying  it  to  Fig.  112,  there  is  obtained  for  the  line  DqD,  passing  through 
the  point  of  intersection  of  the  two  line  polygons,  moments  of  equal  size  for  the 
right  and  left  areas.    In  other  words,  the 
centroidal   axis  and   the    neutral    axis  both 
pass  through  the  point  of  intersection  of 
both  line  polygons. 

To  determine  the   stresses   according  to 
the  formula 

vM 

the   moment   of  inertia  /  of   the  modified 
section  about  the  centroidal  line  is  required.  Fig.  113. 

For  irregular  areas  it    must  be  determined 

graphically.  According  to  the  method  given  by  Mohr,*  the  moment  of  inertia 
in  this  case  is 


/  =  2i7Xarea  ADB, 


so  that  all  the  quantities  required  for  the  computation  of  the  stress  are  known. 

*  In  Fig.  113,  the  moment  of  inertia  of  the  forces  is  equal  to  the  area  enclosed  by  the  line 
polygon,  the  axis  of  inertia  and  the  first  external  side  of  the  line  polygon,  multiplied  by  2iJ, 


132 


CONCRETE-STEEL  CONSTRUCTION 


Applied  to  a  rectangular  cross-section,  a  straight  line  is  obtained  for  the  line 
polygon  starting  from  A,  and  a  parabola  for  that  starting  from  B.  The  com- 
putation of  their  point  of  intersection  leads  again  to  the  equation  of  the  second 
degree,  already  given.    Further,  with  H  =  i  and  b  =  i,  the  distance 


and 


so  that 


2 


_  (h—x)x^ 


xM 
Gh  =  -—r-  = 


I  x^  I ,  x\ 
-X-  =  -   ), 


2M 


Consequently,  the  same  value  is  obtained  as  with  &  =  i,  in  the  formula  previously 
given. 

{b)  Bending  with  Axial  Compression.  First  Method.  The  point  of  applica- 
tion of  the  normal  force  N  is  supposed  to  act  at  C  on  the  axis  of  symmetry 


Fig.  114. — (According  to  Autenrieth.) 

(Fig.  114).  In  a  similar  manner  as  above  described,  the  two  line  polygons  AD 
and  BDG  are  drawn,  wherein  the  latter  also  includes  some  steel  w^hich  would 
])e  within  the  zone  of  compression.*  In  distinction  from  the  case  of  pure  flexure, 
the  neutral  axis  is  shifted  from  D{)  to  Gq.  If  the  distance  of  any  area  element 
of  the  modified  cross-section  from  the  neutral  axis  through  G'o  i^  designated  ^', 
the  following  conditions  of  equahty  are  obtained: 


N  =  i:aXdF  =  -K^dFXv. 

V 

(Equation  for  vertical  component) 


*  If  the  force  N  did  not  exist. — Trans. 


BENDING  WITH  AXIAL  FORCES 


133 


Na^:::dFXov^'^:::dFXv^. 
(Moment  equation  about  neutral  axis) 

Through  a  combination  of  the  two  there  results 

V  V 

from  which 

J'  here  designates  the  moment  of  inertia  of  the  modified  section  and  AI'  its 
statical  moment,  with  reference  to  the  neutral  axis  sought.  Both  quantities  can 
be  represented  graphically  by  using  the  curves  previously  drawn. 

Now 

/'  =  2i7Xarea  ABDGK, 

or  designating  the  area  by /, 


Further 

M'  =  HxKG  =  Hz, 

so  that 

2/ 


or 


z 


az  ^ 

2 


This  equation  provides  a  method  of  locating  the  neutral  axis.  By  laying 
off  from  the  line  AB  (Fig.  114)  the  ordinate  differences  2  between  the  curves  AD 
and  BG,  the  curve  DqO,  starting  from  Do  is  obtained,  and  the  two  shaded  areas 
will  be  equal.    If  DqL  is  made  of  such  size  that  the  triangle  Do^^^^area  ABD 

(ZZ* 

=  triangle  DqOC,  it  follows  that  —  (that  is,  the  area  of  the  triangle  GoCO)  is 
almost  equal  to  the  area  /.    It  is  actually  | 


too  large  by  the  area  enclosed  between 
the  arc  and  the  chord  DqO.  The  po- 
sition Go  of  the  neutral  axis,  therefore, 
still  requires  a  slight  correction.    If  (Fig. 


115)  such  a  piece  COO'  is  cut  off  from  ! 

the  triangle  GqOC,  starting  from  C,  that  yig.  115. 

its  area  equals  that  bounded  by  the  arc 

and  chord  ODq^  then  the  neutral  axis  sought  will  pass  through  0\  because  the 

2/ 

*  Find  z  =  j~:,  by  measurement. — Trans. 


134 


CONCRETE-STEEL  CONSTRUCTION 


area  /  has  lost  only  the  strip  GqGq  O'O,  so  that  the  quantity   ^  is 

equal  to  the  new  area  /. 

After  the  position  of  the  neutral  axis  has  been  determined  in  this  manner, 
the  normal  stress  o  at  any  desired  point  in  the  section  can  finally  be  found. 


_   vN  _vN_vN 

or  also 

V  N  a    V  N  a 


J'         2Hf  ' 

For  the  tensile  stress  in  the  reinforcement,  there  is  found 

nv  N' 
Hz 

The  distances  v  and  a,  as  well  as  z,  are  to  be  determined  from  the  corrected 
position  of  the  neutral  axis. 

Second  Method.^  The  following  is  a  simpler  method  than  that  of  Mohr,  for 
ascertaining  the  unit  stresses  in  a  homogeneous  section  subjected  to  bending  loads 
outside  the  section  itself,  and  in  which  tension  is  excluded,  such  as  may  be 
adopted  for  reinforced  concrete  columns. 

If  the  point  of  application  G  of  the  normal  force  N  lies  on  the  axis  of  sym- 
metry of  the  section,  at  a  distance  a  from  the  neutral  axis,  then,  as  before, 

r 


where  J'  is  the  moment  of  inertia  of  the  modified  section,  and  M'  is  the  statical 
moment  about  the  neutral  axis  being  sought.  In  Fig.  ii6,  the  Hne  polygon  A'B'A" 
is  so  drawn  for  the  force  polygon  on  the  left,  with  a  polar  distance  H,  that  the 
portion  A'B'  belongs  to  the  n-ioldi  steel  section,  while  the  portion  B'A"  is  for 
the  strips  of  the  concrete  section.  If  GK  is  the  true  position  of  the  neutral 
axis,  then  the  statical  moment  of  the  effective  modified  area,  that  is,  of  the  w-fold 
increased  steel  areas  and  the  concrete  area  lying  to  the  right  of  the  axis,  is 

M'=Hz. 

Then,  in  the  line  polygon  of  the  effective  section  CA'  is  the  first  external  side, 
and  the  side  through  G  is  the  last  external  side. 

*  C.  Guidi,  '  'Sul  calcolo  delle  sezioni  in  beton  armato."    Cemento,  1906,  No.  i. 


BENDING  WITH  AXIAL  FORCES 


13.5 


The  moment  of  inertia  of  the  effective  modified  section  is 
/'-2i7Xarea  A'B'GK; 

so  that 

J'     2 X area  A'B'GK 


and 


Fig.  ii6. 


Now,  —  is  also  equal  to  the  area  of  the  triangle  C'GK.    Hence,  necessarily 

CGK^  A'B'GK. 


This  is  the  case  when  the  two  shaded  areas  are  equal. 

To  locate  the  point  G  it  is  necessary  to  draw  from  the  point  C,  located  under 


136 


CONCRETE-STEEL  CONSTRUCTION 


C  on  the  first  external  side  of  the  line  polygon,  a  straight  line  C'G,  so  that  the 
shaded  areas  equal  each  other;  that  is,  so  that 

C'LG=^A'B'L. 

Since  the  area  of  the  figures  A'B'L  is  known,  the  point  G  can  be  easily  and 
exactly  determined,  if  the  difference  is  computed  which  a  slight  displacement 
makes  in  the  value  of  the  shaded  area,  derived  from  first  locating  the  point 
tentatively  according  to  judgment. 

When  the  location  of  the  axis  has  been  fixed,  the  unit  stresses  may  be  com- 
puted with  the  aid  of  the  formulas  of  the  first  method, 


vN 


and 


vN  vN 


HdFXv    M'  Hz' 


V  being  the  distance  from  the  neutral  axis  to  the  extreme  layer. 

The  second  method  seems  somewhat  plainer  than  the  first.  If  desired,  the 
steel  found  within  the  compression  side  can  be  ignored,  and  then  in  the  line 
polygon  B'A'\  simply  the  steel  section  can  be  treated,  as  shown  in  Fig.  117. 


T 

1 

J 

A 

•1 

1 

\\ 

1 

1 

W 

m 

Fig.  117. 

This  second  method  is  very  simple  when  applied  to  the  rectangular  section 
there  shown.  The  line  polygon  A'B'  is  simply  a  straight  line,  and  B'A"  is  a 
parabola. 

It  should  be  noted  that  the  graphical  methods  under  {a)  and  {h)  can  aLc 
be  applied  to  those  forms  of  reinforcement  in  which  the  dimensions  in  a  direction 


BENDING  WITH  AXIAL  FORCES 


137 


parallel  to  that  in  which  the  forces  act  are  so  large  that  the  moment  of  inertia 
of  the  steel  section  must  be  included.  The  section  of  the  reinforcement  is  then 
to  ])e  considered  as  a  concrete  area  composed  of  narrow  strips  ??-times  as  wide 
in  a  direction  parallel  with  the  neutral  axis,  in  order  to  construct  the  line  polygon 

These  larger  sections  of  reinforcement  are  T,  I  and  j^-bars  such  as  are  used, 
for  instance,  in  the  Melan  system. 


METHOD  OF  COMPUTATION  FOR  STAGE  Ila 


For  the  sake  of  completeness,  and  in  order  to  gain  some  insight  into  the 
difficulties  attending  the  exact  analytical  inquiry  as  to  deformations,  the  following 
method  of  computation  for  rectangular  sections  for  Stage  Ila  is  given: 

From  Figs.  92  to  94  on  pages  100  and  loi,  it  is  seen  that  the  curves  of  stress 
in  Stage  II  for  rectangular  reinforced-concrete 
sections,  can  be  closely  approximated  by  two 
straight  lines,  one  of  which  passes  through  the 
neutral  axis  of  the  section  and  is  prolonged  into 
the  tension  side  until  the  tensile  stress  reaches 
a  value  equal  to  the  tensile  strength  of  the 
concrete,  from  which  point  it  becomes  parallel 
with  the  line  representing  the  cross-section. 

With  the  nomenclature  of  Fig.  118,  for 
simple  flexure, 

D^Ze-\-Zb, 

or 

77       ,  v 

2 


Zy- 


haz 


XOz 

a= — ; 

Ob 

whence  Zb=  i  d—x — —^^^boz,  so  that  since  Oe^nob^— — 

2Gb  /  X 


hob      FenobOi—x)  ,  /,  xoz\, 
— X  =  ^          -\-[d—x  boz. 

2  X  \  2(76/ 


For  a  given  value  of  the  ratio 


Ob 


X  may  be  determined  as  follows, 

bx  (Ji—x) 


—  =  Fen'"  "'  +  [d 
2  X 


or 


138 


concrete-stp:el  construction 


whence 


Fefl 


The  location  of  the  neutral  axis  is  thus  determined.    Further,  then, 


M  =  Feae\ 


(i     ^\     7    .7      s/2       d—x\     hoz  ,o  IN 

.  h  -\-h(7z{d—x)  —x-\  a{^x-\-la), 

\      3/  \3         2   /      2  • 

y  =y  +?Oo{d^x){~  +  '^^  -^^aoxfi{2X  +  xd), 


whence 

6Mx 

(7b  = 


nFe{h -x)  {6h - 2x)  -^xhp(d-x)  (3^? +x) -hp^x^{2  +/?) 


If  the  formulas  are  applied  to  the  test  specimen  described  on  page  99  with 
0.4%  of  reinforcement,  and  on  the  assumption  that  w  =  io,  and  for  a  breadth 
of  15  cm.,  the  following  values  are  obtained: 


With/?  =  J 

^  =  15X3659 
cm-kg 


With/9=^i 
^  =  15X5326 
cm-kg 


X 

=  11.4  cm. 

Ob 

^28.0  kg/cm^ 

Oz 

=  9.3  kg/cm2 

Oe 

=  376  kg/cm^ 

X 

=  9.02  cm. 

Ob 

=49.7  kg/cm2 

Oz 

=  9.97  kg/cm2 

Oe 

=  978  kg/cm^ 

measured 


measured 


X  = 

12.4  cm. 

Ob  = 

28.5  kg/cm^ 

Oz  = 

9.5  kg/cm2 

Oe  = 

288  kg/cm2 

X  = 

9.6  cm. 

Ob  = 

48.3  kg/cm2 

Oz  = 

9.5  kg/cm2 

Oe^ 

842  kg/cm2 

With  the  exception  of  o^  the  results  agree  in  a  satisfactory  manner.  It  is 
seen  that  the  computed  position  of  the  neutral  axis  changes  with  an  increase  of 

bending  moment,  and  consequently  the  decreasing  ratio  —  =/9  affects  its  position. 

Ob 

The  limiting  value  with  /?  =  o  corresponds  with  the  computations  for  Stage  lib. 
In  a  beam  loaded  in  the  customary  manner,  so  that  the  bending  moment  increases 
towards  the  center,  the  locus  of  the  neutral  axes  through  the  various  sections 
rises  toward  the  middle  of  the  beam.  At  the  instant  when  cracks  appear,  it  will 
have  reached  a  culminating  point,  which  will  be  lower  in  proportion  to  the 
average  position  of  the  line,  the  higher  are  the  stresses. 


CHAPTER  IX 

THEORY  OF  REINFORCED  CONCRETE 

EFFECTS  OF  SHEARING  FORCES 

While  in  rectangular  steel  beams  the  shearing  stresses  play  small  part  and 
need  be  computed  only  in  exceptional  cases,  in  reinforced  concrete  beams  they 
are  of  considerable  importance  and  must  be  considered  in  the  arrangement  of 
the  reinforcement.  In  reinforced  T-beams  in  which  only  straight  rods  are 
employed,  when  bending  takes  place  (provided  the  reinforcement  is  strong 
enough),  the  break  does  not  occur  near  the  center  of  the  beam  through  tensile 


Fig.  119. — Failure  cracks  in  the  vicinity  of  the  points  of  support  of  a  concrete  T-beam  reinforced 

with  only  straight  round  rods. 

stresses,  but  near  the  points  of  support  where  inclined  cracks  form,  due  to  the 
shearing  stresses  or  the  diagonal  principal  ones  generated  by  them.  Such  cracks 
are  shown  in  Figs.  119  and  120. 

In  homogeneous  beams  possessing  a  constant  modulus  of  elasticity,  the 
diagonal  principal  stresses,  that  is,  the  maximum  values  of  the  tensile  and  com- 
pressive stresses  in  any  inclined  elemental  area,  are  given  by  the  formulas 


0  o 
2      \  4 


140 


CONCRETE-STEEL  CONSTRUCTION 


and  their  direction  by 

2r 

tan  20:  =  —  . 

a 

The  expression 

^  4 

represents  the  limiting  value  of  the  shearing  stress.  The  elemental  areas  in 
which  the  tensile  and  compressive  stresses  act,  and  in  which  the  shear  is  zero, 


Fig.  I20. — Failure  cracks  in  the  vicinity  of  the  points  of  support  of  a  T-beam  reinforced  with 

straight  Thatcher  bars. 

make  with  those  in  which  the  maximum  shearing  stresses  act,  an  angle  of  45°. 
If,  from  point  to  point  of  a  beam,  the  direction  of  the  greatest  (or  least)  principal 
stresses  at  those  points  be  followed,  two  series  of  mutually  perpendicular  curves 
are  traced,  which  are  called  trajectories  of  the  principal  stresses. 

In  Fig.  121  is  given  a  diagram  of  the  trajectories  of  the  principal  stresses  for 
a  simple,  freely  supported,  homogeneous  beam  of  T-section.  All  curves  cut 
the  neutral  axis  at  45°,  at  which  point  g  =  o  and  a^  =  TQ.  If  the  tensile  strength 
is  less  than  the  shearing  strength,  as  is  the  case  for  concrete,  then  the  break 
will  occur  in  consequence  of  the  tensile  stress  a^,  and  the  real  shearing  strength  will 
not  be  developed. 

However,  it  cannot  be  finally  determined  that  the  best  form  of  reinforce- 
ment is  that  which  will  follow  the  direction  of  the  trajectory  of  the  principal 
tensile  stress.  This  point  becomes  evident  upon  working  out  this  idea,  espe- 
cially with  regard  to  continuous  beams  with  variable  loads,  in  which  the  dis- 
tribution of  stress  in  a  section  is  different  in  a  reinforced  from  a  homogeneous 


EFFECTS  OF  SHEARING  FORCES 


141 


beam.  The  principal  stresses  are  also  intluenced  by  vertical  pressures  between 
the  various  separate  concrete  layers. 

At  all  points  in  a  beam  where  0^  =  0,  as  at  the  supports  of  simple  ones  and  at 
the  points  of  zero  moment  of  continuous  ones,  (1=45°.  At  these  points  the 
reinforcement  should  be  bent  at  a  45°  angle  if  it  is  to  conform  to  the  conditions, 
so  that  it  can  take  up  to  the  best  advantage  the  diagonal  tension  stresses,  which 
are  then  equal  to  r.  As  the  middle  is  approached  a,  however,  becomes  smaller 
than  45°,  so  that  flatter  bends  are  advisable  down  to  30°. 

In  adopting  Stage  I  as  a  basis  of  computation,  the  value  of  the  shearing 
stress  r  is  an  approximation  for  the  section  with  0^  =  0. 

In  the  ^'Leitsatze"  and  the  Regulations,"  it  is  required  that  the  horizontal 
tensile  stress  a,  of  the  concrete  is  to  be  wholly  carried  by  the  lower  reinforcement, 
so  that  in  the  calculation  of  the  shearing  stresses  the  tension  in  the  concrete  is 
wholly  ignored.  The  diagonal  tensile  stresses  produced  by  the  shearing  stresses 
should  be  carried  by  stirrups  and  bent  rods.  Since  actual  structures  and  test 
specimens  built  in  this  way  have  proven  satisfactory,  the  simple  method  of  com- 


Fig.  121. — Stress  trajectories  in  a  homogeneous  T-beam. 

puting  the  shearing  stresses  according  to  Stage  lib  will  be  adopted,  and  an 
explanation  given  of  the  various  formulas,  followed  by  a  careful  application  to 
the  experimental  data  at  hand,  in  order  to  ascertain  what  factor  of  safety  is 
provided  against  a  failure  in -the  shearing  strength.  Finally,  will  be  considered 
Considere's  theory  of  the  great  extensibility  of  reinforced  concrete,  under  the 
influence  of  which  the  "  Leitsatze"  was  prepared,  but  which  has  been  found 
untenable  in  practice  and  has  received  certain  modifications. 

FORMULAS  FOR  SHEARING  AND  ADHESIVE  STRESSES 

In  the  same  way  in  which  the  tensile  strength  of  concrete  is  ignored  in  bend- 
ing, the  formulas  for  shear  and  adhesion  will  be  derived  on  the  assumption  that 
the  stresses  a^,  and  are  equally  effective  in  cracked  sections  as  in  all  others. 
Further,  only  plain  reinforcement  will  be  considered. 

I.  Rectangular  section  with  simple  plain  reinforcement  on  the  tension  side. 

The  normal  stresses  are  to  be  found  for  Stage  lib  according  to  method  No.  3 
of  page  80.  Let  AB  and  A'B'  be  two  adjacent  sections  between  which  on  the 
plane  CC  are  applied  shearing  stresses  equal  in  amount  to  the  difference  between 
the  normal  stresses  on        and  A'C  Then 


TXbXdl=  I  bXdvXda. 

J  V 


142  CONCRETE-STEEL  CONSTRUCTION 

It  has  already  been  shown  (page  8i)  that 

2M 


Gh 


bih 


from  which 


dab 
"dl 


hxih 


dM 


x\  dl 


2Q 


bx  h-- 


wherein  Q  represents  the  total  of  the  external  forces  acting  on  one  side  of  the 
section.    From  the  diagram  of  Fig.  122, 


do  —  —dot. 

X 


X  C 

 ^  Q. 

so  that 


B  B' 


TbXdl 


Fig.  122. 


:  r  bXdv 

J  V 


X- 


2vQXdl 
bxmi-- 


xb 


2(3 


x^ih 


/vXdv, 
V 


zb 


Q{x^-v^) 
x^[h-fl 

Consequently,  on  the  top  layer  where  v=x,  the  shearing  stress  is  zero  and 
increases  toward  the  neutral  axis  to 

Q 


h--\b 


The  expressions  for  r  b  and  tq  may  also  be  ol^tained,  if  in  the  regular  formula 
for  homogeneous  sections, 

-  QS 


EFFECTS  OF  SHEARING  FORCES 


143 


is  substituted  the  modified  section  consisting  of  the  compressed  concrete  and  the 
«-times  increased  steel  area.  For  the  computation  of  zq,  the  value  5  of  the 
statical  moment  of  the  compressed  concrete  with  reference  to  the  zero  axis  is 

2 

and  the  moment  of  inertia  is 

J^^bx^-\-nFe(h-xf, 

so  that 

2 

T0  = 


It  follows,  however,  from  the  quadratic  ecjuation  for  the  determination  of 
that 


2nh—x 


.0=  ^ 


bih-- 


so  that  finally  as  before 


3 

According  to  the  assumptions  made  for  Stage  116,  no  normal  stress  acts  in 
the  concrete  below  the  neutral  axis,  the  whole  tensile  forces  being  taken  by  the 
reinforcement.  With  this  supposition,  the  shearing  stress  tq  is  constant  between 
the  line  00'  and  the  reinforcement.  In  that  case  it  is  evident  that  the  shearing 
stress  To  is  also  ec^ual  to  the  difference  in  the  tensile  stress  between  two  adjacent 
sections  of  the  reinforcement. 


Hence  it  follows  that 


bToXdl=dZ; 
3 

dZ    dM      I  Q 


dl      dl  I x\  x'' 
'    3/     '  3 


so  that 


_Q_ 
3 


144 


CONCRETP]-STEEL  CONSTRUCTION 


This  value  of  bzo  also  represents  the  total  effective  adhesive  stress  on  unit 
length  of  the  circumference  of  the  steel,  and  consequendy  the  adhesive  stress  ri  is 

bzo 


total  circumference  of  the  reinforcement* 

Example.  A  reinforced  concrete  slab  with  ^^  =  6.79  cm^  (i.o5in2)=6  rods 
12  mm.  (J  in.  approx.)  in  diameter,  has  a  span  of  2.0  m.  (6.56  ft.)  and  carries 
a  load  of  820  kg/m^  (168  lbs/ ft^).  In  it  //=9cm.  (3.54  in.).  The  distance 
of  the  neutral  axis  from  the  upper  layer,  computed  according  to  formulas  already 
given,  is 

^  =  3-38  cm.  (1.33  in.). 
Further,  for  &  =  ioo  cm.  (39.4  in.), 

(2  =  820  kg.  (1704  lbs.). 

Hence 

iooTo  =  -^-^^^  =  io4  kg/cm2  (1479  lbs/in2), 

To  =  i.04  kg/cm^  (14-8  lbs/in^). 

Since  the  reinforcement  per  meter  width  consists  of  6  rods  12  mm.  in  diameter, 
the  total  circumference  is 

^  =  6X3.14X1.2  =  22.6  cm.  (8.9  in.), 

and  the  adhesive  stress 

Ti=^=4.6  kg/cm2  (65.4  lbs/in2). 

In  simple  slabs  the  shearing  and  adhesive  stresses  are  usually  so  small  that 
their  computation  seems  unnecessary.  For  the  same  reasons,  stirrups  in  simple 
slabs  are  deemed  superfluous. 

Of  more  importance  are  the  shearing  and  adhesive  stresses  in 

2.  T-beams.    It  is  evident  that  the  expression  in  the  last  section 


which  applies  to  rectangular  sections,  is  also  applicable  to  T-sections  when  the 
distance  of  the  reinforcement  from  the  centroid  of  compression  (Fig.  loi,  page 


EFFECT«  OF  SHEARING  FORCES 


145 


X 

io8)  is  substituted  for  //  ,  and  for  b  the  width  of  the  stem  /;o  is  used.  For 

3 

x<d  the  expression  remains  the  same,  while  for  x>d  (Fig.   102,  page  109), 


is  to  be  used.    Approximately,  the  somewhat  too  small  value  h  may  be 

2 

used,  so  that  for  the  distance  of  the  centroid  of  compression  from  the  reinforce- 
ment, 

Q 

^0- 


(which  is  slightly  too  large)  is  the  shearing  stress  in  the  stem  between  the 
reinforcement  and  the  neutral  plane. 

ExafHple.    For  the  freely  supported  T-beam  of  Example  i,  page  11  l, 

/  =  5.5  m.  (18.0  ft.), 

5  =  3780  kg/m  (2535  lbs/running  ft.). 


Thus, 


(3  =  2.75X3780  =  10,395  kg.  (22,869  lbs.), 


^0*0  =  ^^37^9^  =  ^96  kg/cm, 


and  the  shearing  stress  in  the  stem  is 

To  =  ^  =  7.o  kg/cm2  (99.6  lbs/in2). 

If  all  five  of  the  28  mm.  (ij  in.  approx.)  rods  were  carried  to  the  support, 
the  adhesive  stress  at  that  point  would  be 

  =4.5  kg/ cm^  (64.0  lbs/ in-). 


5X3-14X2 


The  foregoing  formulas  for  the  shearing  stress  are  deduced  on  the  assumption 
that  no  tensile  stresses  act  on  the  concrete  below  the  neutral  axis.  It  is  to  be 
noted,  however,  that  in  both  Stage  I  and  lla,  where  the  concrete  yet  carries 

some  tension,  the  compressive  force  D  =  — ,  where  z  is  the  distance  between 

z 

Z  and  D.    Furthermore,  the  horizontal  shearing  force  at  the  level  of  the  neutral 


146 


CONCRETE-STEEL  CONSTRUCTION 


axis  between  two  adjacent  sections  must  carry  the  whole  of  D,  so  that,  since  the 
distance  z  is  constant  between  two  successive  sections,  it  follows  that 

dD  Q 

hTQ=—r=  —  . 

dl  z 


The  difference  which  exists  between  the  actual  value  of  the  shearing  stress 
in  the  neutral  layer  compared  with  the  assumption  made  on  the  basis  of  Stage 
11^,  will  only  be  caused  by  the  difference  between  the  calculated  lever  arm  between 
the  centroids  of  tension  and  compression,  and  the  actual  distance.  From  the 
column  giving  the  value  of  y  in  the  table  on  page  99,  it  is  plain  that  these 
values  do  not  differ  much  in  rectangular  sections,  and  an  examination  of  the  stress 
distribution  shown  in  Figs.  92  to  94  proves  that  the  actual  distance  is  somewhat 

X 

smaller  than  that  computed  from  the  expression  h  .    In  consequence,  in 

rectangular  sections,  the  value  tq  along  the  neutral  layer  is  slightly  greater  than 
that  given  by  computations  on  the  basis  of  Stage  11^.  In  T-beams,  when 
ignoring  the  effect  of  the  tensile  stresses  in  the  concrete  of  the  stem,  the  result- 
ant Z  of  the  tensile  stresses  falls  nearer  the  steel  stress  Z^,  and  the  arm  of  the 
couple  between  tension  and  compression  in  the  section  will  be  somewhat  larger 
according  to  the  computations  than  in  reality.  It  is  then  to  be  expected  that 
in  T-beams,  because  of  the  influence  of  the  slab,  the  shearing  stresses  in  the  stem 
will  actually  be  more  closely  given  by  the  approximate  formula 

^  Q 

Later,  the  relation  of  shear  to  reinforcement  will  be  taken  up.  An  exact 
theoretical  method,  however,  is  seen  to  be  very  difficult  of  development,  in  view 
of  the  uncertainty  of  computations  based  on  Stage  l\a,  when  it  is  to  be  con- 
sidered that  the  stress  distribution  shown  in  Fig.  118  corresponds  only  approx- 
imately with  fact. 

The  adhesive  stresses  given  by  the  formulas  developed  above  for  Stage  l\h 
are  too  large  when  compared  with  the  actual  conditions  at  the  appearance  of 
the  first  tension  crack,  and  for  Stages  I  and  Ila.  According  to  the  table  on 
page  99,  the  value  of  Z^  increases  up  to  the  appearance  of  the  first  crack. 
Consequently,  the  increase  of  Z^,  which  is  directly  proportional  to  the  adhesive 
stress,  is  slower  with  increase  of  external  moment  than  that  of  D,  on  which  the 
computation  of  ri  is  based.  In  T-beams,  where  the  influence  of  Zj,  is  less, 
the  agreement  will  be  better  between  the  computed  Zh  and  the  actual  than  in 
rectangular  sections. 

The  shearing  force  h^TQ,  found  along  the  neutral  axis,  also  acts  in  large  part 
along  the  planes  aa'  perpendicular  to  it,  which  form  the  connection  between 
the  stem  and  the  slab  (Fig.  123).  The  average  value  of  the  shearing  stress  at 
those  points  will  be 

^_5oTo  h—hp 
2d  h 


EFFECTS  OF  SHEARING  FORCES 


147 


In  the  planes  aa'  there  is  no  lack  of  reinforcement,  since  the  slab  rods  are 
there  present  in  consideraljle  numbers.  Their  shearing  resistance,  however, 
does  not  come  into  play  in  taking  their  share  of  the  transfer  of  the  shearing 
stresses,  but  rather,  their  better  tensile  qualities  are  active.    If  it  be  imagined 


w  ^  

a.           a  \ 

d 

i  . 

•  • 

1 

a' 

Fig.  123. 


that  the  left  flange  of  the  T-beam  is  cut  away  along  the  plane  aa' ,  besides  the 
shearing  stresses  r,  others  perpendicular  to  them  and  due  to  the  bending  of 
the  slab,  will  be  brought  into  action  on  the  section.  This  bending  will  be 
resisted  by  the  combination  of  one  flange  with  the  stem  and  the  slab  on  its 
opposite  side,  so  that  tensile  and  compressive  stresses  normal  to  the  plane  are 
brought  into  play  to  counteract  the  bending  which  would  be  produced  by  the 
shearing  stresses  r,  and  the  flange  is  held  in  its  actual  condition,  under  stress. 
(Fig.  124.)    To  these  tensile  stresses  are  added  the  bending  tensile  ones  in  the 


r 

r 

1 

1 

: 

cc 


Fig.  124. — Distribution  of  shearing  and    Fig.  125. — Probable  courses  of  the  stress  trajectories 
normal  stresses  along  the  section  aa' .        in  a  floor  slab  acting  as  a  compression  member. 


slab  itself,  due  to  the  moment  at  the  support.  All  the  stresses  described  above 
produce  tension  and  compression  trajectories  in  the  slab,  which  take  somewhat 
the  courses  shown  in  Fig.  125.  Thus,  the  slab  reinforcement  lies  so  as  to  be 
favorable  to  the  production  of  a  reduction  in  the  principal  tensile  stresses,  which 
are  here  less  than  those  of  shear,  since  the  accompanying  compressive  stresses 
diminish  their  amount.    If  the  tensile  stresses  are  entirely  annulled,  the  com- 


148 


CONCRETE-STEEL  CONSTRUCTION 


pressive  trajectories  will  become  arch  lines,  which  will  be  held  in  balance  hy 
the  tensile  strength  of  the  slab  reinforcement.  If  the  beams  are  close  together 
the  arches  may  overlap  one  another. 

If  there  act  on  the  sides  of  an  infinitesimal  parallelopiped  (Fig.  126)  the 
pairs  of  shearing  stresses  t,  and  also  the  mutually  perpendicular  normal  stresses 

and  Oy,  then  the  values  of  the  principal  stresses  may  be  computed  by  the 
formulas 

and  their  direction  by 

2T 

tan  2a  =  . 

Ox  —  Oy 


Fig.  126. 


In  the  case  in  hand,  since  and  Oy  cannot  be  determined  with  certainty,  an 
exact  theoretical  treatment  of  the  question  as  to  the  distribution  of  the  stresses 

in  a  flat  plate  is  very  difficult,  and  without  checking 
by  experiments  (which  are  still  rare)  would  be 
worthless.  Moreover,  it  is  evident  that  the  round- 
ing or  sloping  of  the  joint  between  slabs  and  stems 
of  beams  is  of  considerable  value  in  the  transfer 
of  the  shearing  stresses  at  those  points.  Although 
in  practice  this  point  is  often  ignored,  and  the 
shearing  stresses  r  along  the  plane  aa'  are  really 
excessive  in  many  actual  structures  (although 
dangerous  consequences  have  not  yet  de- 
veloped), it  is  invariably  best  to  follow  only 
accepted  and  safe  methods. 
The  ends  of  reinforcing  rods  should  always  be  made  with  a  hook  so  that  sole 
dependence  is  not  placed  on  friction  or  adhesion.  For  this  purpose  the  shape 
of  the  hook  is  of  importance.  The  form  commonly  employed,  of  a  simple 
right-angle  bend,  is  not  very  effective  when  surrounded  only  by  a  thin  concrete 
slab,  as  is  often  the  case  at  the  ends  of 
beams.  In  such  cases  the  ends  should 
rather  be  given  a  larger  bend  of  as 
much  as  90°.  Considere,  in  the  French 
section  of  the  International  Society  for 
Testing  Materials,  reported  a  new  form 
of  the  end  hook,  which  should  be  immedi- 
ately adopted  in  practice.    By  bending 

the  end  into  a  half  circle,  through  which  a  short  straight  piece  may  be  fastened, 
the  principle  of  rope  friction  is  employed  and  a  greater  frictional  resistance  is 
produced  on  the  inner  side  of  the  bend,  since  the  hook  will  be  pressed  hard 
against  the  concrete.  Some  expriments  by  Considere  led  to  the  result  that  rods 
with  ends  bent  into  semicircles  could  be  stressed  to  the  elastic  limit,  while  the 
adhesion  of  plain  rods  is  between  13.4  and  24.3  kg/cm^  (191  and  346  lbs/in^). 


Fig.  127. — Form  of  hook,  according  to 
Considere. 


EFFECTS  OF  SHEARING  FORCES 


149 


When  the  average  unit  resistance  to  sliding  developed  by  these  hooks  is  computed, 
it  is  found  to  be  about  77.4  kg/ cm^  (1095  lbs/in^)  of  contact  between  the  steel 
and  concrete,  or  about  three  times  that  of  plain  rods. 

These  hooks  possess  the  further  merit  of  not  depending  to  any  great  extent 
upon  the  character  of  the  concrete  or  the  care  given  the  work,  since  a  rope- 
like friction  is  secured  by  the  large  curve  of  pressure.  This  pressure  naturally 
should  not  be  too  large,  since  then  a  crushing  of  the  concrete  results.  According 
to  Considcre,  the  best  results  are  secured  by  giving  the  semicircular  bend  a 
diameter  about  five  times  that  of  the  rod. 

The  Action  of  Stirrups 

In  the  special  literature  of  this  subject  the  opinion  is  generally  advanced  that 
vertical  stirrups  have  the  power  of  reducing  the  shearing  (schub-)  stresses  in 
the  concrete  because  of  their  shearing  strength,  that  they  are  stressed  in  shear 
as  well  as  in  tension.  In  order  to  compute  the  distribution  of  the  shearing  stress 
between  the  concrete  and  the  stirrups,  the  area  of  the  latter  is  to  be  considered 


Fig.  128.  Fig.  129. 

as  mcreased  n-fold.  The  weakness  of  this  idea  is  proven  by  the  following  points: 
If  a  piece  of  length  dh  be  imagined  as  cut  from  a  stirrup  stressed  thus  in  shear 
(the  section  of  which,  for  sake  of  simphcity,  is  assumed  as  square)  (Fig.  128), 
then  it  can  be  in  equilibrium  under  the  action  of  the  shearing  stresses  r^a^  acting 
at  the  ends  of  the  section,  only  when  another  couple  due  to  adhesion  comes  into 
action.  Then 

rea^Xdh  =  TiaXdhXa  +  2TiX-XdhX- 

2  2 

must  follow,  so  that  r^  =  i.5ri.  That  is,  the  shear  in  a  stirrup  cannot  exceed  one 
and  a  half  times  the  adhesive  strength.    Similarly,  for  circular  stirrups 


150 


CONCRETE-STEEL  CONSTRUCTION 


The  normal  stresses  upon  the  sides  of  the  stirrup  sections  are  infinitely  small 
quantities  of  the  second  order,  and  are  not  considered.  Also,  normal  stresses 
within  the  section  itself  of  a  stirrup  cannot  assist  in  producing  equilibrium, 
because  then  the  bending  stresses  in  two  adjacent  sections  must  l)e  opposite  in 
sign  (Fig.  129)  according  to  this  theory.  Round  stirrups  can  thus  be  stressed 
in  shear  to  a  maximum  which  is  scarcely  more  than  their  full  adhesive  strength, 
which  latter  is  practically  nothing  compared  with  their  observed  efficiency. 
An  allowable  shear  for  purposes  of  computation,  no  larger  than  the  allowable 
adhesion  is  worthless.  The  favorable  influence  of  the  stirrups  in  the  following 
experiments  can  be  explained  only  through  their  acting  in  tension. 


CHAPTER  X 


THEORY  OF  REINFORCED  CONCRETE 

EXPERIMENTS  CONCERINQ  THE  ACTION  OF  SHEARING 

FORCES 

The  following  results  were  secured  by  the  author  near  the  end  of  1906, 
from  experiments  on  T-beams.  The  accuracy  of  the  conclusions  drawn  from 
them  can  be  checked  by  means  of  the  now  well-known  experiments  of  the 
Eisenbetonkommission  der  Jubilaumstiftung  der  Deutschen  Industrie,  in  the 
preparation  of  the  outline  of  the  program  of  which  the  author  assisted  as  a 
member. 

Experiments  by  the  Author.  The  experiments  were  not  conducted  on  T-beams 
designed  according  to  normal  methods,  but  such  dimensions  were  chosen  as 
would  cause  failure  by  exceeding  either  the  value  ti  of  the  adhesion,  or  tq,  that 
of  the  shear  in  the  rib.    The  two  beams  were  joined  by  a  continuous  slab,  so 


•  0,60  i 

i«  X].SO  4.  ■  uo  > 

1'  

1  i 

«  0 

j 
1 

i 

r  : 

1 

M 

Fig.  130. — Section  of  test  beams. 


that  the  load,  which  consisted  of  bars  of  iron  and  sacks  of  sand,  could  be  uni- 
formly applied  and  not  produce  torsional  stresses  as  is  possible  with  a  single  beam 
and  slab.  Roofing  felt  was  applied  to  the  ribs  over  the  supports,  so  as  to  reduce 
friction.*  The  slab  was  so  strongly  built  that  it  would  carry  with  safety  the 
breaking  loads.    (See  Fig.  130.) 

The  small  span  of  2.70  mm.  (8.86  ft.)  was  adopted,  so  that  the  relation 
between  the  reactions  and  the  center  moments  would  be  out  of  proportion;  in 
other  words,  so  that  failure  would  take  place  at  the  ends  before  the  middle 
broke.  The  two  beams  of  each  specimen  were  similarly  reinforced.  The  scheme 
of  loading  the  twelve  specimens  involved  three  groups,  in  the  first  of  which  the 
load  was  uniformly  distributed;   in  the  second,  two  symmetrically  placed  con- 

*  In  the  majority  of  cases  the  friction  at  the  supports  was  eliminated  by  supporting  one 
end  from  a  windlass  so  that  it  was  free  to  oscillate. 

151 


152 


CONCRETE-STEEL  CONSTRUCTION 


centrated  loads  were  used;  and  in  the  third,  the  beam  was  broken  by  a  single 
center  load.  The  specimens  were  about  three  months  old,  the  concrete  was 
mixed  in  the  proportions  of  one  part  Heidelberg  Portland  cement  to  4J  parts 
Rhine  sand  and  gravel,  of  such  size  that  there  were  72  parts  of  sand  of  o  to  7  mm. 
(o  to  ^  in.)  grains,  and  28  parts  of  pebbles  of  7  to  20  mm.  to  J  in.)  diameter. 
The  sides  and  bottoms  of  the  beams  were  whitwashed  so  as  better  to  reveal  the 
cracks.  Without  such  a  white  coating  the  first  cracks  could  not  be  found  till 
a  much  later  period. 

The  six  beams  of  the  first  group  for  uniformly  distributed  loading,  had  the 
same  quantity  of  steel  in  each  beam,  although  variously  distributed. 

Beam  I.  Three  straight  round  bars  18  mm.  (J  in.  approx.)  in  diameter 
with  ends  hooked;  one-half  of  the  beam  without  stirrups,  and  the  other  half 
with  them. 

Beam  II.    The  same  as  I,  except  that  the  ribs  were  twice  as  broad. 

Beam  III.  Three  straight  Thacher  rods  without  hooks;  one-half  of  the 
beam  without  stirrups,  the  other  half  with  them. 

Beam  IV.  The  same  area  of  reinforcement  as  I  and  II,  except  that  there 
was  one  rod  of  18  mm.  (J  in.  approx.)  diameter,  and  three  round  rods  of  15  mm. 
{yq  in.  approx.)  diameter,  the  latter  bent  upward  at  an  angle  of  45°  near  the  sup- 
ports, the  straight  rod  hooked  at  the  end,  the  whole  beam  without  stirrups. 

Beam  V.  The  same  area  of  reinforcement  in  the  form  of  two  rods  16  mm. 
(f  in.),  and  two  rods  15  mm.  in.  approx.)  in  diameter,  the  latter  bent  in  the 
form  of  a  truss  from  the  third  points  to  the  tops  of  the  beams  over  the  supports. 

Beam  VI.  Like  IV,  except  with  stirrups  throughout  the  whole  length  of 
the  beam,  the  lower  straight  rod  without  hooks. 

The  loadings  produced  the  following  results  in  the  several  beams: 

Beam  I  (Fig.  131).  Three  straight  round  rods  of  18  mm.  (f  in.  approx.) 
diameter,  with  hooks  at  the  ends.  For  a  total  load  of  11.5  t.  (12.68  tons) 
on  both  beams,  the  computed  stresses  according  to  the  ^^Leitsatze"  were  (7^  = 
loookg/cm^  (14223  lbs/in^)  on  the  steel,  ^7^  =  17.8  kg/cm^  (253  lbs/ in^)  com- 
pression in  the  concrete,  To  =  8.4kg/cm2  (119  lbs/in^)  shear  over  the  supports,  and 
Ti=6.g  kg/cm^  (98  lbs/in^)  adhesion.  Thus,  with  this  otherwise  permissible 
load,  the  shear  in  the  ribs  was  excessive. 

At  a  load  of  7.0  t.  (7.7  tons)  the  first  cracks  appeared  (fine  tension  ones  near 
the  center),  corresponding  to  a  computed  stress  of  (7^  =  668  kg/cm^  (9501  lbs/in^). 
The  computation,  according  to  Stage  I,  with  ^  =  15,  gave  (7^  =  22.7  kg/cm^  (321 
lbs/ in-).  With  increasing  loads  the  tension  cracks  became  more  numerous  and 
larger,  and  on  the  end  of  the  beam  with  stirrups,  some  followed  the  stress 
trajectories.  When  the  load  reached  15  t.  (16.5  tons)  there  appeared  on  the 
left  side,  that  is,  the  end  without  stirrups,  a  distinct  inclined  crack,  which, 
starting  from  the  top  gradually  extended  to  the  steel.  At  this  load  <Tg  =  i26o 
kg/cm^  (17,921  lbs/in^)  at  the  center,  and  to  =  io.5  (213  lbs/in^),  and  71=8.65 
kg/cm^  (123  lbs/in^)  at  the  ends  of  the  beams. 

The  further  failure  of  the  beam  took  place  with  increased  load  by  an  exten- 
sion of  the  inclined  crack  on  the  left  just  above  the  lower  reinforcement  along 
it  to  the  support,  so  that  the  steel  was  thrown  into  compression  and  the  adhesion 
between  it  and  the  concrete  of  the  rib  was  lost.    At  a  load  of  25.7  t.  (28.3  tons) 


154 


CONCRETE-STEEL  CONSTRUCTION 


the  horizontal  crack  had  extended  entirely  to  the  end  of  the  rod,  so  that  the 
whole  of  the  force  acting  on  it  was  carried  into  the  concrete  through  the  hook. 
It  was  clearly  observed  that  under  this  load  the  horizontal  and  diagonal  cracks 
enlarged,  the  hook  finally  straightened  out  and,  because  of  the  high  compression, 
the  concrete  cracked,  and  failure  followed  suddenly. 


Fig.  134. — Beam  I,  cracks  in  the  end  without  stirrups  at  the  breaking  load. 

Under  the  load  of  25.7  t.  (28.3  tons)  the  right  end  of  the  beam,  which  was 
provided  with  stirrups,  also  showed  a  diagonal  crack  at  a  point  corresponding 
almost  exactly  with  the  one  which  caused  the  break  at  the  other  end.  The 
computed  stresses  at  rupture  were  =  38.0  (540  lbs/ in^),  0^  =  2060  (29,300  lbs/in^) 
in  the  center,  and  to  =  i6.9  (240  lbs/in-)  and  ti  =  13.9  kg/cm^  (198  lbs/in^)  at 
the  supports.    The  last  two  quantities  naturally  apply  only  to  the  practically 


Fig.  135. — Beam  I,  cracks  in  the  end  suppHed  with  stirrups,  at  the  breaking  load. 

uninjured  right  end  of  the  beam.  Figs.  134  and  135  give  characteristic  views 
of  the  behavior  of  the  two  differently  reinforced  ends.* 

Beam  II  (Fig.  132)  differed  from  the  foregoing  one  only  in  the  double  width 
of  the  ribs.  The  first  very  fine  tension  crack  appeared  near  the  center  at  a 
load  of  13.7  t.  (15.1  tons)  and  corresponded  to  a  computed  steel  stress  of  0^  = 
1200  kg/ cm-  (17,067  lbs/in^).    (For  Stage  I  with  ^  =  15,  0^  =  26.^  kg/cm^  (381 

*  The  cracks  were  blackened  so  that  they  would  show  in  the  photograph,  and,  except  when 
plainly  recognizable  as  failure  cracks,  were  much  finer  than  the  lines  seen. 


ACTION  OF  SHEARING  FORCES 


155 


lbs/ in-).  Thus,  the  first  tension  crack  did  not  appear  at  the  expected  load  of 
11.5  t.  (12.7  tons).  This  is  to  be  ascribed  to  the  action  of  the  concrete  in  the 
tension  zone,  so  that  the  reduction  of  stress  produced  by  it  in  the  steel,  which 
influences  the  formation  of  tension  cracks,  will  make  the  suggested  method  of 
computation  give  somewhat  higher  results  for  a  width  of  rib  of  28  cm.  than  for 
one  only  14  cm.  wide.  With  increase  of  load  the  tension  cracks  increased  in 
number  and  at  a  steel  stress  of  r7^,  =  i5oo  kg/cm^  (21,335  lbs/in-)  were  quite 
cons])icuous.  At  a  load  of  30  t.  (33  tons)  two  diagonal  cracks  appeared  at  the 
left,  one  of  which  extended  upward  along  the  under  side  of  the  slab  and  down- 
ward along  the  reinforcement,  until  finally  at  a  load  of  40  t.  (44  tons)  failure 
occurred  through  widening  of  these  cracks,  and  a  pulling  out  of  the  reinforce- 
ment over  the  support  followed.    The  computed  stresses  were 

at  30  t  (33  tons),  rT,  =  24io  (34,279),  to  =  io  (142),  ti  =  i6.5  kg/cm2  (235  lbs/in^), 
at  40  t  (44  tons),  (7^  =  3150  (44,804),  To  =  12.9  (183),  T]  =21.2  kg/cm^  (301  lbs/in^). 

Beam  III  (Fig.  133).  The  reinforcement  consisted  of  three  straight  Thacher 
rods  without  hooks.  This  kind  of  reinforcement  is  theoretically  of  value  in 
increasing  the  adhesion,  which  is  here  really  not  in  question.  If  failure  had 
occurred  in  the  first  two  beams  through  lack  of  adhesion,  the  failure  in  this  case 
should  be  different.  The  Thacher  rods  were  not  of  constant  section,  the  round 
part  havin  g2.54  cm-  (0.394  in-)  and  the  flattened  portion  2.04  cm^  (0.372  in^)  area. 

At  a  load  of  6.8  t.  (7.48  tons)  the  first  tension  cracks  appeared  in  the 
neighborhood  of  the  center,  corresponding  to  a  computed  steel  stress  of  a^  = 
710  kg/cm^  (10,090  lbs/in^).  When  the  load  reached  13  t.  (14.3  tons)  other 
later  vertical  cracks  had  appeared,  and  also  the  first  diagonal  cracks  near  the 
supports.  To  this  load  corresponded  computed  stresses  of  o-^  =  1370  kg/cm^ 
(19,486  lbs/ in-)  in  the  center,  and  70  =  9.3  (171  lbs/in^),  71  =  7.25  kg/cm^  (103 
lbs/in^)  at  the  supports.  At  17.6  t.  (19.4  tons)  this  crack  had  extended  both 
upward  and  downward,  and  also  in  the  end  supplied  with  stirrups  a  diagonal 
crack  was  visible.  In  Fig.  136  is  clearly  seen  the  separation  between  the  steel 
and  its  covering,  promoted  by  the  spreading  effect  of  the  "  knots."  Failure 
resulted  at  a  load  of  19.5  t.  (21.5  tons)  with  corresponding  values  of  ^^,  =  1960 
kg/cm2  (27,877  lbs/in^),  -0  =  13.2  (198  lbs/in^),  and  71  =  10.3  kg/cm^  (147  lbs/in^). 

At  the  end  containing  stirrups,  the  bursting  effect  was  not  seen  in  the  con- 
crete. Just  how  great  was  the  effect  of  these  stirrups  cannot  be  determined  from 
this  experiment. 

If  the  causes  and  the  formation  of  the  cracks  in  these  three  beams  are 
examxined,  it  is  established  that  the  cracks  first  became  visible  where  the  moment 
was  greatest,  and  that  with  increase  of  load  other  more  distant  cracks  appeared. 
On  the  end  supplied  with  stirrups,  the  cracks  appeared  to  occur  at  the  sections 
in  which  the  stirrups  were  located,  since  the  concrete  section  was  weakened  at 
those  points. 

In  a  uniformly  loaded  beam,  when  the  first  tension  cracks  occur  in  the  middle  and 
penetrate  to  the  reinforcement,  then  naturally  the  tensile  strength  of  the  concrete 
is  no  longer  effective  and  the  steel  must  carry  the  whole  tension  in  the  cracked 


156 


CONCRETE-STEEL  CONSTRUCTION 


sections.  The  concrete  will  tend  to  contract  slightly  on  each  side  of  a  crack,  while 
the  steel  will  be  stretched  more  at  such  a  point,  and  the  consequence  will  be  that 
the  two  will  move  in  opposite  directions  until  the  frictional  resistance  has  decreased 
the  stress  in  the  steel  and  increased  that  in  the  concrete  so  that  both  materials  are 


1, 


Fig.  136, — Beam  III,  with  Thacher  rods  under  the  breaking  load,  at  the  end  without  stirrups. 


stretched  an  equal  amount.  The  breadth  of  the  crack  measured  directly  along 
the  steel,  thus  shows  the  amount  of  the  slip  of  the  concrete  over  the  steel  in  a  certain 
length. 

On  the  concrete  of  a  rib  between  two  cracks  must  act  the  difference  JZ  of  the 
tensions  in  the  steel  at  the  two  cracks,  this  difference  being  the  total  frictional  re- 
sistance between  steel  and  concrete  of  the  corresponding  length.  Further,  the 
concrete  will  exert  bending  stresses  on  the  reinforcing-rod,  as  shown  in  Fig.  137, 

   which  counteract  the  deformation 

J  ^  in  the  piece   of   concrete,  which 

would  be  caused  by  JZ.  These 
stresses  must  exist  because  of  the 
somewhat  inclined  positions  of  the 
cracks.    One  condition  favorable  to 
the  stresses  which  act  in  a  piece  of 
concrete  between  two  cracks  is  that 
the  section  which  experiences  the 
first  crack  is  stressed  to  a  higher 
point  than  the  others,  the  centroid 
of  pressure  of  the  former  lying  higher,  while  the  position  of  the  centroid  of  tension 
in  the  reinforcement  does  not  vary  with  the  increasing  length  of  the  cracks. 
Consequently,  the  decrease  JZ  of  the  tension  precedes  that  of  the  bending 


Fig. 


^37- 


ACTION  OF  SHEARING  FORCES 


157 


moment.  This  is  of  particular  importance  near  the  centers  of  uniformly  loaded 
beams  in  which  a  sort  of  arch  action  takes  place. 

The  cracks  near  the  supports  (which  were  clearly  inclined  in  direction),  and 
which  led  to  Imal  rupture,  are  to  be  clearly  distinguished  from  those  near  the  mid- 
dles of  the  beams,  which  commenced  low  and  extended  upward  with  increase  of 
load.  In  technical  literature,  the  idea  has  been  advanced  in  the  effort  to  obtain 
much  lower  unit  adhesive  stresses,  that  these  diagonal  cracks  producing  failure 
are  due  to  an  overcoming  of  the  adhesion  between  the  steel  and  concrete.  It  is 
believed,  however,  that  these  first  three  experiments  disclose  the  weakness  of  that 
idea.  If  exceeding  the  adhesive  strength  really  was  the  cause  of  the  cracks,  a])prox- 
imately  equal  values  of  ri  should  be  obtained  at  the  load  at  which  the  cracks  first 
appeared  in  Beams  I  and  II.  They  differ  considerably,  however,  since  the  two 
values  are  71=8.65  and  16.5  kg/cm^  (123  and  235  lbs/in^).  In  Beam  III,  in  which 
the  adhesive  strength  was  not  in  question,  the  diagonal  crack  occurred  earlier  than 
in  Beam  I.  It  is  thus  seen  that  the  diagonal  cracks,  which  may  lead  to  failure, 
start  at  loads  which  are  proportional  to  the  breadths  of  the  beams,  and  the  con- 
clusion is  justified  that  the  tensile  strength  of  the  ribs  in  a  diagonal  direction  was 
exceeded,  and  that  in  this  case  shearing  stresses  were  primarily  involved. 

When  such  a  diagonal  crack  exists,  no  diagonal  tensile  stresses  can  act  in  the 
concrete  at  the  faces  of  such  crack  and  the  left  hand  portion  (see  Fig.  138)  is  held 


f  f  r  r 


Fig.  138. 


LJL 


in  equilibrium  by  the  lateral  force  Q,  the  steel  tension  Z,  and  the  force  D,  of  the 
arch  compression  of  the  concrete.  These  three  forces  must  intersect  at  a  common 
point,  and  consequently  D  and  Z  are  not  parallel,  as  is  shown  in  the  diagram. 
Since  the  Z  on  the  left  side  of  the  crack  acts  in  a  direction  inclined  downward  to 
the  right,  it  is  evident  that  the  turning  of  the  two  parts  of  the  beam  resulting  from 
the  opening  of  the  cracks  will  cause  the  reinforcement  on  the  right  end  of  the  left 
hand  part  of  the  beam  to  press  downward  (see  Fig.  139),  so  that  at  that  point  in 


Fig.  139. 


the  steel  a  downward  force  will  act  which  naturally  cannot  be  larger  than  the  ten- 
sile strength  between  the  concrete  beneath  the  reinforcement  and  that  in  the  rib 
above  it.    The  narrower  is  the  breadth  of  the  concrete  around  the  rods,  under 


158. 


CONCRETE-STEEL  CONSTRUCTION 


otherwise  similar  conditions,  the  sooner  must  the  diagonal  crack  become  horizontal 
over  the  reinforcement,  and  it  is  evident  that  both  cracks  might  appear  at  the  same 
time.  In  Beam  I,  3  t.  (3.3  tons)  was  the  increment  of  the  load  necessary  to  extend 
the  crack  horizontally;  and  in  Beam  II  it  was  7  t.  (7.7  tons);  wliile  the  total  loads 
at  that  time  were  18  and  37  t.  respectively  (19.8  and  40.7  tons). 

When,  then,  the  connection  between  the  steel  and  the  concrete  is  destroyed  by 
the  downward  pressure,  the  adhesion  is  no  longer  effective  and  the  adhesive  strength 
is  of  no  further  avail.  The  tension  Z  will  then  become  constant  along  the  rod  to 
the  hook,  and  failure  must  occur  when  the  hook  cannot  stand  the  pull.  The 
longer  is  the  horizontal  crack  along  the  reinforcement,  the  more  nearly  horizontal 
will  Z  act,  and  the  more  inclined  will  D  become,  since  both  forces  must  intersect  Q, 
and  the  consequence  is  that  the  slab  is  lifted  away  from  the  rib  on  the  right,  this 
action  being  promoted  by  the  effect  on  the  bending  stress  of  the  adhesion  between 
the  steel  and  the  portion  of  the  concrete  rib  at  the  right.  That  is  the  explanation 
of  the  extension  of  the  break  horizontally  between  the  slab  and  the  rib,  as  a  con- 
tinuation of  a  diagonal  crack. 

The  excess  of  the  breaking  load  of  Beam  II  over  that  of  I  can  be  explained  by 
the  fact  that  the  hooks  secured  a  better  hold  in  the  broader  rib  than  in  the  narrower 
one. 

From  these  descriptions  a  conception  may  be  obtained  of  the  action  oi  vertical 
stirrups.  It  must  be  assumed  that  even  in  the  presence  of  stirrrups,  similar  stresses 
exist  in  the  concrete  of  the  rib  as  when  none  are  present,  since  the  cracks  in  the 
vicinity  of  the  support  on  the  side  supplied  with  stirrups  were  also  clearly  inclined. 
The  action  of  the  stirrups  may  then  be  considered  such  that  in  a  diagonal  section 
the  diagonal  tension  of  the  concrete  and  the  major  forces  Z  and  D,  together  with 
the  tensions  of  the  stirrups,  hold  in  equilibrium  the  external  forces  on  the  left  side 
of  the  section.  Fig.  140.    First,  the  diagonal  tension  in  the  concrete  will  be  less 

and  the  diagonal  cracks  will  occur  later. 
Further,  the  downward  pressure  of  the 
lower  reinforcing  rods,  through  exceeding 
the  diagonal  tensile  strength  of  the  con- 
crete, will  be  prevented,  and  also  the 
splitting  apart  of  the  slab  and  the  rib. 
Thus,  the  stirrups  counteract  a  rupture 
over  a  considerable  distance  near  the 
end  of  a  beam,  and  in  this  respect  the 
increase  of  the  adhesive  strength  on  the  end  supplied  with  stirrups  can  be  ascribed 
to  their  influence.  As  to  just  how  far  the  ability  of  the  stirrups  extends  in  this 
regard,  the  previous  experiments  give  no  information,  and  other  beams  must  be 
tested  which  are  provided  with  stirrups  throughout  their  full  length. 

In  Beams  I  and  II  the  location  of  the  diagonal  crack  which  eventually  produced 
failure,  may  apparently  be  determined  from  the  position  of  ihe  section  where  the 
upper  end  of  the  crack  met  the  under  side  of  the  slab,  which  section  experienced 
with  the  corresponding  loading,  the  same  bending  moment  as  did  the  center  of  the 
beam  at  the  occurrence  of  the  first  tension  crack.  In  the  first  two  experiments  the 
upper  ends  of  the  cracks  were  located  40  cm.  (15.8  in.)  from  the  supports,  and  the 
corresponding  bending  moments  computed  in  this  way  are  1.45  and  2.75  m-t.,  while 


r  f  r 


Fig.  140. 


ACTION  OF  SHEARING  FORCES 


159 


the  corresponding  center  moments  at  the  appearance  of  the  first  tension  cracks 
were  1.51  and  2.70  m-t. 

If  the  values  of  To  in  these  sections  are  computed  at  the  appearance  of  the  diag- 
onal cracks,  there  result,  for 

Beam  I,    70  =  -^^"^'^^ 7'^  kg/cm^  (105.3  lbs/in2), 
Beam  II,  70=^—^-^— =  7.0  kg/cm^  (99.6  lbs/in^), 

which  are  in  close  accord  with  the  directly  measured  tensile  strength  of  7.7  kg/cm^ 
(109.5  lbs/in2). 

It  is  thus  demonstarted  that  at  the  neutral  axis  the  shearing  stress  develops  the 
tensile  strength  of  the  concrete  and  acts  at  an  angle  of  45°  with  the  neutral  plane. 
In  consequence  of  the  presence  of  large  tensile  stresses  in  their  vicinity,  cracks  will 
first  tend  to  occur  near  the  points  of  support.  But  the  shearing  stresses  will  be 
predominant  at  those  points,  since  with  the  first  relative  movement  between 
concrete  and  reinforcement,  the  normal  tensile  stresses  will  be  diminished.  This 
explains  the  rapid  extending  of  the  diagonal  "shearing"  cracks,  which  quickly 
reached  from  their  point  of  origination  to  the  under  side  of  the  slabs,  thus  reach- 
ing higher  than  the  earlier  tension  cracks  which  formed  in  the  central  portions  of 
the  beams. 

Another  question  remains:  Why  are  not  the  diagonal  cracks,  which  arise 
mainly  from  shearing  stresses,  most  numerous  near  the  supports  where  the  total 
force  is  greatest?  To  this,  two  explanations  may  be  advanced.  First,  in  the 
vicinity  of  the  supports,  vertical  compressive  stresses  have  to  be  resisted,  arising 
from  the  reaction  of  the  support,  which  diminish  the  principal  stresses.  Con- 
sequently, the  hypothesis  of  loading  of  Stage  lib  does  not  apply.  Computed 
according  to  Stage  I,  the  maximum  shearing  stress  -  at  the  neutral  axis  is  some- 
what larger,  but  it  rapidly  diminshes  upward  and  downward,  so  that  a  reduction 
can  readily  be  imagined  as  taking  place  in  that  vicinity. 

The  favorable  influence  of  the  stirrups  is  obvious  in  Figs.  131  to  133.  Cracks 
also  formed  where  the  stirrups  existed,  most  inclined  near  the  supports,  but  they 
did  not  open  as  widely  as  in  the  other  halves.  According  to  Fig.  140,  the  stirrups 
which  are  supposed  cut  by  a  plane  at  an  angle  of  45°,  will  prevent  a  premature 
failure  at  the  end  of  the  beam,  when  they  are  collectively  able  to  resist  the  total 
lateral  force,  so  that  Z  and  D  may  act  horizontally.  Under  this  supposition,  the 
stirrups  at  the  end  cracks  would  be  stressed  in  Beam  I  to  (7  =  2900  kg/cm-  (41,248 
lbs/in^)  and  in  Beam  II  to  (7  =  3900  kg/cm-  (55,471  lbs/in^).  Manifestly,  the 
stirrups  lying  near  the  ends  resist  the  downward  pressure  of  the  reinforc- 
ing rods. 

Under  the  assumptions  here  made,  the  stirrups  act  as  the  vertical  tension 
members  of  a  truss,  while  the  compression  members  are  diagonals  inclined 
toward  the  center.  (Fig.  141.)  Where  the  cracks  rise  nearly  vertically,  the 
forces  are  correspondingly  small,  and  at  those  points  of  the  ribs  the  stirrups  act 


160 


CONCRETE-STEEL  CONSTRUCTION 


also  as  reinforcement  against  the  bending  produced  by  the  sliding  resistance  of 
the  reinforcing  rods. 

Beam  IV  (Fig.  147).    The  reinforcement  consisted  of  three  round  rods  15  mm. 

in.  approximately)  in  diameter,  and  one  rod  18  mm.  (f  in.  approximately) 
in  diameter,  thus  being  just  as  large  as  in  the  preceding  specimens.  Of  these 
four  rods  the  three  df  15  mm.  diameter  were  bent  upward  at  an  angle  of  45°  at 
the  points  w^here  the  moment  diagram  allow^s  it.  With  a  load  of  11.5  t.  (12.7  tons) 
the  computed  stresses  were:  (7e  =  iooo  kg/cm^  (14,223  lbs/in^),  (76  =  18.9  kg/cm^ 
(269  lbs/in^).  To  =  8.5  kg/cm^  (121  lbs/in^),  and  for  the  adhesion  the  value 
Ti  =  2i.i  kg/cm^  (300  lbs/in^)  when  only  the  single,  lower  straight  rod  is  con- 
sidered, with  Ti  =  6.o  kg/cm^  (85  lbs/in^),  if  all  the  steel  is  considered.  If  the 
normal  tensile  stresses  in  the  concrete  which  produce  the  diagonal  tensile  ones 
in  connection  with  the  shearing  stresses  are  imagined  as  resisted  by  the  bent 
portions  of  the  lower  reinforcing  rods,  then,  according  to  Fig.  142,  all  the  area 


Fig.  141.  Fig.  142. 

elements  which  slope  at  an  angle  of  45°  toward  the  center  are  in  tension  and  are 
to  be  summed,  while  the  opposite  ones  are  in  compression,  which  the  concrete 
easily  withstands.  The  diagonal  tension,  which  is  nothing  at  the  center  under  a 
uniform  load,  can  be  represented  by  the  area  of  a  trapezoid,  the  smaller  base 
of  which  corresponds  with  a  tensile  stress  of  to  =  2.o  (28  lbs/in^)  for  concrete 
with  a  factor  of  safety  of  four.  On  this  assumption  (which  may  be  justified  in 
the  case  of  great  extensibility  of  the  concrete),  the  bending  upward  of  the  rods 
is  to  be  done  so  that  they  pass  through  the  centroids  of  the  three  equivalent 
trapezoids  making  up  the  large  one.  In  this  instance  the  total  tension  to  be 
carried  by  the  three  rods  is 

2  =  ^:^1^X73X14=5366  kg.  (11,805  lbs.), 
2 

so  that  their  unit  stress  is 

(7g=:5^— =approx.  1000  kg/cm^  (14,223  lbs/in^). 
5  *3 

Reinforced  concrete  beams  can  also  be  considered  as  trusses  with  single  or 
double  web  systems  (Figs.  143  and  144),  wherein  the  diagonal  concrete  layers 


ACTION  OF  SHEARING  FORCES 


161 


represent  the  compression  meml)ers.  The  tensile  stresses  in  the  bent  rods  can  be 
checked  equally  well  from  the  forces  acting  in  the  direction  of  the  diagonal,  or  from 
the  theory  that  they  are  the  members  of  a  single  or  more  complex  intersection 
system,  or  from  the  shearing  stress  tq. 

Since  in  the  foregoing  experiments  I-III,  the  cracks  near  the  supports  had 
an  inclination  of  approximately  45°,  it  is  doubtful  whether  the  steel  intersecting 
them  was  stressed  simply  in  tension.  Consequently,  the  adhesion  should  be  com- 
puted differently  from  the  method  of  p.  146,  where  only  straight  rods  were 


Fig.  143- 


Fig.  144. 


considered.  If  the  truss  arrangement  is  assumed,  a  constant  stress  must  exist 
in  the  lower  rods  from  the  support  to  the  last  bend,  which  stress  can  be  computed 
from  the  moment  of  the  section  through  the  corresponding  joint  of  the  truss. 
In  a  double  intersection  system,  this  joint  is  the  point  of  intersection  of  the  first 
diagonals;  in  a  single  system  it  is  the  first  top  joint.  Both  fall  close  to  the 
support,  so  that  the  moment  may  practically  be  derived  from  the  reaction  at 
that  point.  If  this  is  represented  by  Q,  and  ignoring  for  the  sake  of  simplicity 
all  load  between  the  support  and  the  first  joint,  then,  on  the  assumption  of 
diagonals  at  an  angle  of  45°,  the  tension  Z  in  the  first  lower  chord 


of  a  double  intersection  system  is  Z 
of  a  single  system  is 


2  _Q. 


Z  ^ 


This  tension  must  be  transferred  to  the  concrete  up  to  the  next  point  of  bend, 
by  the  end  hook  and  the  adhesion.    A  properly  constructed  J-formed  hook 


Fig.  145- 


would  fully  and  safely  transfer  the  stress;  but  if  it  is  desired  to  compute  the 
adhesive  stress  on  the  basis  of  no  hook  action,  then,  if  the  adhesion  be  con- 
sidered as  uniformly  distributed  over  the  first  panel  length  of  the  rods,  if  U  is  the 
circumference  of  the  straight  rods 


in  a  double  intersection  system,  ri  = 
in  a  single  system,  n  = 


_Q_. 

2ZU' 

_Q_ 

2ZU' 


162 


CONCRETE-STEEL  CONSTRUCTION 


a  value  only  half  that  found  on  the  assumption  of  straight  rods  only.  These 
formulas  apply  only  when  the  diagonal  tension  (  =  to)  is  all  carried  by  the  bent 
reinforcing  rods,  and  the  other  method  is  to  be  employed  when  the  value  for  ti 
will  cause  sliding,  as  determined  by  the  experiments  on  beams  of  corresponding 
construction.  So  far  as  the  foregoing  experiments  allow  of  a  decision,  the  values 
of  the  adhesion  found  by  the  formulas  correspond  very  well  with  the  figures  for 
direct  shding  resistance. 

The  action  of  Beam  IV  under  load  will  next  be  discussed.  With  9  t.  (9.9  tons) 
(see  Fig.  147),  the  first  tension  crack  appeared  near  the  center  corresponding  to 
a  stress  of  Oc  =  Sio  kg/cm^  (11,521  lbs/in^).  According  to  Stage  I  with  ^  =  15, 
the  tension  in  the  concrete  was  (7,  =  2'j.i  kg/cm^  (385  lbs/in^).  At  the  same 
time,  at  the  supports  the  shearing  stress  79  =  7.0  kg/cm^  (100  lbs/in^)  and  the 
adhesion,  according  to  the  formula  derived  above,  was  ti=S.j  kg/cm^  (125  lbs/in^). 
Other  tension  cracks  appeared  at  13.8  (15.2),  14  (15.4),  and  18  t.  (iq.8  tons). 
The  diagonal  cracks  next  the  ends  appeared  it  a  load  of  33  t.  (36.3  tons),  the 
one  at  the  left  causing  failure  at  42  t.  (46.2  tons).  At  33  t.  (36.3  tons)  it  is 
computed  that  70  =  21.7  (309  lbs.)  and  -1  =  26.8  kg/cm^  (385  lbs/in-),  and  it  is 
seen  that  since  the  adhesion  would  exceed  its  usual  maximum  value  with  increase 
of  load,  the  nearest  bent  rods  must  have  carried  considerable  stress.  If  the 
assumption  of  a  truss  action  is  made,  the  beam  corresponds  with  one  with 
inchned  end  posts,  and  the  first  bent  section  thus  carried  a  doubly  great  stress 
Consequently,  the  bent  rods  exerted  a  bursting  action  on  the  concrete,  which 
would  ultimately  cause  failure.  The  computed  stresses  at  rupture  were  (7^  =  62 
(882  lbs.),  cre  =  326o  (46,368  lbs.),  7:0  =  27  (384  lbs.),  and  ti  =33.5  kg/cm^  (476 
lbs/in^),  the  latter  figure  evidently  having  no  practical  meaning.  This  experi- 
ment proves  that  it  is  important  so  to  arrange  the  lower  rods  which  run  con- 
tinuously to  the  ends,  that  they  cannot  he  pulled  out,  and,  that  a  good  round  hook  is 
important  at  the  ends  of  bent  rods. 

In  connection  with  Beam  IV  will  be  described: 

Beam  VI  (Fig.  148),  in  which  the  main  reinforcement  was  arranged 
like  that  of  Beam  IV,  except  that  the  straight  rod  had  no  hooks  and  was 
carried  beyond  the  end  of  the  concrete,  so  that  the  _  first  slip  could  be 
observed.  Stirrups  were  supplied  throughout  the  whole  length  of  the  beam, 
and  the  center  one  of  the  bent  rods  terminated  slightly  short  of  lines  oter  the 
points  of  support. 

The  first  tension  cracks  in  the  middle  w'ere  visible  under  a  load  of  6  t.  (6.6. 
tons),  corresponding  with  a  computed  stress  ^7e  =  590  kg/cm^  (8392  lbs/in^).  The 
first  diagonal  cracks  occurred  at  19  and  20  t.  (20.9  and  22  tons),  commencing 
in  the  tension  side  exactly  like  those  of  Beam  IV.  At  the  same  time  the  rod 
18  mm.  (x^  in.)  in  diameter  which  protruded  from  one  end  had  begun  to  move 
inward.  Then,  70  =  13.1  (186  lbs.)  and  71  =  16.3  kg/cm^  (232  lbs/in^).  Because 
of  the  slipping,  the  stresses  in  the  bent  rods  in  the  vicinity  of  the  supports  were 
augmented. 

Failure  took  place  at  a  load  of  37.8  t.  (41.6  tons),  with  computed  stresses  of 
<^6  =  56  (797  lbs.),  <7,  =  295o  (41,959  lbs.),  70  =  24.5  (348  lbs.),  and  71=30.4  kg/cm2 
(432  lbs/in^).  The  last  figure  naturally  has  no  meaning.  Because  of  the  slipping 
of  the  straight  rod,  the  compression  at  the  bends  of  the  outer  rods  was  increased, 


164 


CONCRETE-STEEL  CONSTRUCTION 


so  that  at  those  points,  as  in  Beam  IV,  the  concrete  covering  was  burst.  (See 
the  photograph  of  Fig.  150.) 

By  a  comparison  with  IV,  it  is  seen  that,  with  a  proper  arrangement  of  the 
main  reinforcing  rods,  the  addition  of  stirrups  has  no  influence  on  the  shearing 
stresses,  but  it  is  important  that  the  lower  rods  which  are  straight  should  be 
fully  secured  against  slipping.  To  this  end  they  should  be  hooked  and  be  in 
such  numbers  as  will  provide  proper  safety  against  sHpping.  The  larger  breaking 
load  found  in  IV  is  entirely  due  to  the  hooks  on  the  ends  of  the  straight  rod. 

Beam  V  (Fig.  149).  The  reinforcement  was  like  that  of  the  Hennebique 
system,  with  two  straight  rods  16  mm.  (f  in.)  in  diameter  and  two  15  mm.  (^  in. 
approximately)  in  diameter,  so  bent  as  to  form  a  suspension  system.  The  last 
two  were  bent  at  the  third  points,  and  extended  as  far  as  the  supports.  One- 
half  of  the  beam  had  no  stirrups,  while  at  the  other  end  were  single  stirrups 
closely  clasping  each  rod. 


Fig.  150. — Beam  VI,  under  the  breaking  load. 


With  a  load  of  7  t.  (7.7  tons)  the  first  tension  cracks  appeared  in  both  beams, 
the  computed  steel  stress  then  being  (7^  =  702  kg/cm^  (9985  lbs/in^).  When  the 
load  had  been  increased  to  ii  t.  (12. i  tons),  at  the  end  without  stirrups  a  diagonal 
crack  appeared  in  the  upper  part  of  one  of  the  ribs,  which  produced  final  failure 
at  31  t.  (34.1  tons).  In  the  other  beam  the  corresponding  crack  first  showed 
itself  at  17  t.  (18.7  tons),  the  difference  being  due  probably  to  unequal  loading. 
If  the  shearing  stress  at  the  upper  ends  of  the  cracks  is  computed  for  the  average 
of  the  two  loads,  tq  is  found  equal  to  7.65  kg/cm^  (109  lbs/in^),  corresponding 
well  with  Beams  I  and  II.  The  development  of  the  failure  was  exactly  like  that 
of  I,  II,  and  III,  in  which  the  turning  of  the  two  parts  about  their  point  of  con- 
tact made  the  lower  reinforcement  at  the  left  of  the  crack  exert  pressure  down- 
ward, while  the  slab  was  cracked  away  from  the  rib  above.  At  the  support,  the 
end  with  stirrups  developed  an  inclined  crack  at  a  load  of  31  t.  (34.1  tons). 
Under  the  breaking  load,  the  computed  stresses  were:  ^6  =  48.3  (687  lbs.),  0^  = 
2600  (36,981  lbs.),  To  =  2i  (299  lbs.),  Ti=29.4  kg/cm^  (418  lbs/ in^), "  the  latter 
computed  from  the  circumference  of  the  two  straight  rods  according  to  the 
''Leitsatze." 


ACTION  OF  SHEARING  FORCES 


165 


From  a  comparison  of  Beam  V  with  I,  II,  and  III,  it  follows  that  with 
uniform  loads,  the  suspended  system  of  reinforcement  does  not  give  any  increase 
of  safety  against  the  appearance  of  diagonal  tension  cracks,  or  the  final  failure 
produced  by  them,  as  compared  with  straight  rods  without  stirrups,  and  that 
stirrups  are  so  much  the  more  necessary.  Beam  V  carried  only  slightly  more 
than  I,  less  than  II,  and  only  three-quarters  of  IV,  so  that  the  superiority  of 
reinforcement  along  the  trajectories  is  clearly  shown.  In  this  connection  it  should 
not  be  forgotten  that  IV  and  VI  were  intentionally  so  designed  that  the  ultimate 
adhesive  strength  of  the  straight  rods  would  be  exceeded. 

The  first  group  of  experiments,  as  already  stated,  gave  no  indication  whether 
the  carrying  power  would  have  been  increased  if  the  stirrups  had  been  carried 
the  whole  length  of  Beams  I,  II,  III, 
and  V.  Furthermore,  the  question  is 
still  open  as  to  whether  a  decrease  of  zi 
by  the  use  of  two  or  three  straight  rods 
in  IV  and  VI  would  have  postponed 
failure  to  any  extent. 

The  three  beams  of  the  second 
group  were  designed  for  two  concentrated 
loads  at  the  third  points,  and  differed  Fig.  151. 

from  those  of  the  first  group  only  in  the 

arrangement  of  the  reinforcement,  which,  however,  was  of  the  same  total  area. 

Beam  VII  (Fig.  152)  had  four  rods  of  16  mm.  (f  in.)  diameter,  one  of  which 
was  carried  straight  to  the  supports  and  was  hooked  at  the  ends,  while  the  others 
were  so  bent  as  to  cut  the  layer  in  which  the  force  was  constant,  so  as  to  divide 
it  equally.  (Fig.  151.)  Stirrups  were  provided  throughout  the  whole  length. 
For  a  safe  load  of  2P=9  t.  (9.9  tons),  the  computed  stresses  were:  g^^toio 
(14,365  lbs.),  ro  =  7.o  (icq  lbs.),  and  ri=9.8kg/cm2  (139  lbs/in-),  according  to 
the  formula 

The  diagonal  tension  brought  onto  one  of  the  bent  rods  was 

Z^-  =  2100  kg  (4620  lbs.), 

1-414X3 

so  that  its  unit  stress  was 

(Te=— —  =  1040  kg/cm^  (14,792  lbs/in^). 


The  first  tension  cracks  were  seen  in  both  beams  under  a  load  of  7.5  t.  (8.3  tons) 
and  were  uniformly  distributed  over  the  central  portion  having  a  constant  bend- 
ing moment.  In  this  condition  the  stress  (7^  =  862  kg/cm-  ([2.261  lbs/in^),  while 
the  tension  in  the  concrete  according  to  Stage  I,  with  n  =  i$j  was  computed  at 
(7^  =  29.2  kg/cm2  (415  lbs/in^).  With  increase  of  load  the  tension  cracks  extended 
upward,  and  near  the  supports  other  cracks  appeared,  corresponding  with  the 


166 


CONCRETE-STEEL  CONSTRUCTION 


diagonal  tensile  stresses  observed  in  connection  with  straight  and  bent  rein- 
forcement. The  final  load  was  slightly  eccentric,  so  that  failure  occurred  on  the 
right  at  34  t.  (37.4  tons).  On  the  assumption  of  a  concentrated  load  of  that 
amount,  the  computed  stresses  are  thus  somewhat  too  small: 

^76  =  65  (925  lbs.),    rT^  =  3420  (48,643  lbs.),    79  =  22.4  (319  lbs.), 
'2^1=31-5  kg/cm2  (448  lbs/in2). 

No  loosening  of  the  ends  of  the  straight  rods  was  observable. 

Beam  VIII  (Fig.  153).  The  reinforcement  consisted  of  four  rods  16  mm. 
(f  in.)  in  diameter,  and  was  arranged  hke  a  suspension  system,  in  which  half  the 
rods  were  bent  directly  from  the  third  points  to  points  over  the  supports.  The 
width  of  the  rib  was  only  10  cm.  (3.9  in.),  and  stirrups  were  used  for  only  one-half 
the  length  of  the  beam. 

It  is  a  widely  held  opinion,  that  in  this  arrangement  of  reinforcement,  intro- 
duced by  Hennebique,  a  part  of  the  load  is  carried  by  the  bent  bars  to  the  supports 
and  that  thus  in  simple  beams  with  the  suspension  form  of  reinforcement,  the 
whole  of  the  reaction  does  not  act  near  the  ends  as  a  shearing  force.  If  this 
suspension  theory  of  the  Hennebique  system  has  any  validity,  it  must  be  verified 
in  this  case,  in  which  the  suspension  rods  have  exactly  the  equilibrium  curve  for 
a  part  of  the  concentrated  loads.  The  first  tension  crack  became  visible  at  a 
load  of  5  t.  (5.5  tons).  To  this  corresponds  a  stress  of  0^  =  648  kg/cm^  (9217 
lbs/in^).  Other  cracks,  distributed  over  the  middle  third,  followed  soon  after.  At 
9.8  t.  (10.8  tons)  a  nearly  horizontal  crack  appeared  above  the  bent  rods  at  the 
left.  At  this  point  tq  is  computed  as  10.7  kg/cm^  (152  lbs/in^),  and  taking  into 
account  the  weight  of  the  beam  which  slightly  increases  the  lateral  forces  at  the 
crack,  -1=9.7  kg/cm^  (138  lbs/in^).  According  to  the  suspension  theory,  about 
half  of  the  load  was  carried  directly  by  the  bent  rods,  so  that  the  other  half  came 
upon  the  plain  beam,  which  then  was  stressed  to  to  =  4.8  kg/cm^  (68  lbs/in^). 
This  does  not  explain  the  horizontal  crack,  however.  At  14  t.  (15.4  tons),  the 
crack  extended  downward  in  an  inclined  direction,  for  which  load  70  =  14.2 
(202  lbs/in^).  It  is  to  be  noted  that  since  the  horizontal  crack  started  at  9.8  t. 
(10.8  tons),  the  suspension  system  actually  carried  about  half  of  the  load,  so  that 
the  To  of  the  plain  beam  amounted  to  approximately  7.1  kg/cm^  (loi  lbs/in^). 

Failure  resulted  from  a  widening  of  the  diagonal  cracks  and  downward 
pressure  of  the  reinforcement  near  the  supports,  at  23.4  t.  (25.7  tons),  for 
which  are  computed:  ^6  =  47. 6  (677),  (7^  =  2450  (34,847),  ro  =  22.8  kg/cm^  (324 
lbs/in^);  the  adhesive  stress  ti  being  as  large  as  tq  if  it  is  computed  for  a  beam 
with  only  two  straight  rods.  The  hooks,  which  had  to  carry  the  whole  of  the 
tension,  burst  the  concrete  at  failure.    (See  Fig.  155.) 

Beam  IX  (Fig.  154)  had  the  same  reinforcement  as  VIII,  but  was  14  cm. 
(5.5  in.)  wide. 

The  first  tension  crack  occurred  at  5.9  t.  (6.5  tons),  with  a  corresponding 
stress  of  ^7^  =  735  kg/cm^  (10,454  lbs/in^).  At  14.5  t.  (16.0  tons)  cracks  appeared 
on  the  beams,  which  indicated  a  looseness  of  the  suspension  rods;  for  this  load 
To  =  9.9  kg/cm^  (141  lbs/in^),  not  quite  so  large  as  for  Beam  VIII.  With  24.5 
t.  (27.0  tons)  in  the  rear  beam  the  diagonal  crack  extended  toward  the  support. 


168 


CONCRETE-STEEL  CONSTRUCTION 


Again  assuming  that  the  suspension  system  carried  half  the  load,  then  for  the 
plain  beam  to  =  8,.j  kg/cm^  (124  lbs/in^),  which  is  practically  equal  to  the  tensile 
strength  of  concrete.  At  failure,  which  took  place  at  25.6  t.  (28.2  tons),  in  a 
manner  similar  to  that  of  Beam  VIII,  the  computed  stresses  were:  (76  =  52.2 
(742  lbs.),  (7^  =  2690  (38,261  lbs.),  ^0  =  17.7(252  lbs.),Ti  =  24.8  kg/cm^  (353  lbs/in^), 
except  that  with  the  suspension  theory  the  last  two  stresses  would  be  only 
half  as  large.  The  stirrups,  supplied  on  one  end,  through  their  tensile  strength, 
hindered  the  formation  of  diagonal  cracks,  and  showed  themselves  essential  and 
indispensable  elements  in  the  Hennebique  system.  The  limit  of  their  effect  is, 
however,  not  disclosed  by  these  experiments.  According  to  the  method  here 
given  of  computing  the  stresses  in  the  stirrups,  they  should  in  this  case  have  been 
stressed  to  1700  kg/cm^  (24,180  lbs/ in^)  at  the  cracking  of  the  concrete.  In 
any  case,  from  the  results  of  the  second  group  of  experiments  can  be  deduced 
the  facts  that  the  bending  of  the  reinforcement  according  to  the  theory  concern- 


FiG.  155. — Beam  VIII,  under  the  breaking  load. 


ing  the  diagonal  tensile  stress  tq  is  much  more  effective  than  according  to  the 
suspension  theory,  in  this  case  the  ultimate  loads  being  in  the  proportion  of 
34:23.4:25.6.  The  reinforcement  of  Beams  VIII  and  IX  demands  stirrups, 
and  it  should  be  established  through  special  experiments  how  much  better  they 
act  than  a  reinforcement  of  simply  straight  rods  and  a  similar  arrangement  of 
stirrups. 

The  third  group  of  specimens  was  designed  for  a  concentrated  load  at  the 
center  of  the  beam.  Since  the  same  amount  of  reinforcement  was  used  as  in 
the  second  group,  it  is  clear  that  the  action  of  the  external  force  w^ould  be  coun- 
terbalanced by  the  resisting  moment  of  the  center  section.  In  fact,  the  failure 
of  Beams  X  to  XII  took  place  through  exceeding  the  tensile  strength  of  the 
reinforcement. 

Beam  X  (Fig.  157).  This  beam  was  reinforced  according  to  the  trajectory 
system,  with  four  rods  of  16  mm.  (f  in.)  diameter,  of  which  three  were  bent  and 


ACTION  OF  SHEARING  FORCES 


169 


one  was  carried  straight  through  to  the  supports;  'one-half  of  the  beam  was 
without,  and  the  other  half  was  supplied  with  stirrups. 

For  the  first  tension  crack,  which  had  extended  well  upward  at  a  load  of  7.5  t. 
(8.3  tons),  the  computed  stresses  were  (T^,  =  i240  (17,637  lbs.),  to  =  6.i  (87  lbs.); 
and  the  adhesion,  according  to  the  formula 


was  Ti  =8.5  kg/cm^  (121  lbs/in^).  With  increase  of  load,  a  large  number  of  tension 
cracks  appeared,  without  disclosing  any  substantial  difference  between  the  two 
halves  of  the  beam.  Failure  took  place  at  27  t.  (29.7  tons)  through  opening  of 
the  center  cracks  and  crushing  of  the  concrete  at  the  upper  side  of  the  slab.  (See 


Fig.  160.)  The  computed  stresses  were:  (7^  =  77.5  (1102  lbs.),  (7^=4050  (57,605 
lbs.),  ro  =  i8.i  (257  lbs.),  ti  =  25.4  kg/cm^  (361  lbs/in^). 

In  this  case  the  computation  of  is  worthless  since  the  pressure  zone  which 
theoretically  should  be  7  cm.  (2,75  in.)  high,  was  reduced  to  about  2  cm.  (0.79  in.) 
by  the  extending  of  the  cracks  upward  to  such  a  considerable  extent  because  of  the 
great  stretch  of  the  steel.  If  a  new  arm  of  30.2  cm.  (11.9  in.)  for  the  couple  be- 
tween tension  and  compression  be  used  to  compute  the  stresses,  with  Z=J9  =  31,300 
kg.  (68,860  lbs.), 

^,  =  ^^-^  =  3880  kg/cm2  (55,187  lbs/in2), 
6.4 

ah  =  ^^^^'^^^  =  2^2  kg/cm^  (3727  lbs/in^). 
120.2 


The  strength  of  compression  cubes  averaged  182  kg/cm^  (2589  lbs/in^). 


ACTION  OF  SHEARING  FORCES 


171 


Beam  XI  (Fig.  158)  contained  reinforcement  in  the  form  of  a  suspension 
system  with  no  stirru])s  whatever.  At  6  t.  (6.6  tons)  the  first  very  fine  tension 
cracks  became  visible,  extending  well  upward  in  the  center,  corresponding  with 
stresses  of  rT^.  =  io45  (14,86311)8.),  70  =  5.3  (75.4  lbs.),  ti -7.42  kg/cm^  (105.5 
lbs/in^).  The  remainder  of  the  jjhenomena  were  exactly  like  those  of  X.  Failure 
occurred  in  the  same  manner  at  26  t.  (28.6  tons),  while  onward  from  22.5  t. 
(24.8  tons),  the  center  cracks  widened  rapidly.  At  failure,  ^7^  =  4000  (56,894  lbs.), 
(7^=83  (1181  lbs.),  70  =  17.9  (255  lbs.),  "1=25.1  kg/cm-  (357  lbs/in-).  Here  also 
Of^  is  to  be  corrected,  as  in  the  last  specimen,  so  that  3800  kg/cm^  (54,049  lbs/ in-) 
steel  stress  is  obtained. 

Beams  XII  (Fig.  159).  The  reinforcement  consisted  of  four  rods  16  mm. 
(f  in.)  in  diameter,  of  which  one  was  bent  up  at  an  angle  of  45°,  the  two  middle 


Fig.  160. — Beam  X,  under  the  breaking  load. 

ones  at  an  angle  of  30°,  while  one  rod  was  carried  straight  through  to  the 
supports.  The  action  under  load  was  exactly  like  that  of  X.  The  first  tension 
crack  occurred  at  5.5  t.  (6.1  tons),  corresponding  to  rj,  =940  kg/cm^  (13,370 
lbs/in^).  At  26  t.  (28.6  tons)  the  ultimate  carrying  capacity  was  exceeded,  at 
com.puted  stresses  of  (76  =  74.6  (1061  lbs.),  (7^  =  3900  (55,471  lbs.),  70  =  17.5  (249  lbs.), 
71  =  24.5  kg/cm^  (348  lbs/ in^).  With  the  actual  lever  arm  of  30.2  cm.  (12.6  in.) 
there  is  given 

Z=D  =  3o,ioo  and  ^7^=— =3740  kg/cm-  (53,195  lbs/in^). 
0.04 

The  beams  of  the  last  group  thus  gave  no  indication  concerning  the  action 
of  the  shearing  forces,  and  to  solve  this  question  experiments  must  be  made  with 
heavier  reinforcement.-  The  stresses  in  the  reinforcement  at  failure  consider- 
ably exceeded  the  elastic  limit  of  the  steel.  Other  experiments  should  be  per- 
formed with  regard  to  this  point  on  beams  with  rather  wide  slabs,  so  that  failure 


172 


CONCRETE-STEEL  CONSTRUCTION 


Table  XXXI 


The  figures  in  heavy  type  are  those  causing  failure. 


Beam  Number. 

First  Crack. 

Commencement  of  Diagonal  Crack  which  led  to  Final  Failure. 

Load  Q  in 

Ge 

az 

Load  Q  in 

Ge 

Tl 

To 

At 
Supports. 

At  Upper 
End  of 
Crack. 

t 

c 

0 

B 

\. 

M 

's 

\ 

m 

B 
0 

\ 

M 

\ 

in 

en 
C 
0 

a 

B 

\ 

B 
0 

M 

B 

\ 

I 
II 
III 

IV 
V 
VI 
VII 
VIII 
IX 
X 
XI 
XII 

7.0 

13-7 
5.8 
9.0 
7.0 
6.0 
7-5 
5-1 
5-9 
7-5 
6.0 

5.5 

7-7 
15-1 
6.4 
9-9 

7-  7 
6.6 

8-  3 
5-6 
6.5 
8-3 
6.6 
6.1 

668 
1 200 
710 
810 
702 

590 
862 
648 

735 
1240 

1045 
940 

9501 
17068 
10098 
II52I 

9985 

8392 
I2261 

9217 

10454 
17637 
14863 
13370 

22.7 
26.8 
19.8 
27.1 
22.  2 
20.  I 
29.  2 

25-1 
24.2 
41  .0 

33-9 
31.6 

323 
381 
282 

385 
316 
286 
415 

357 
344 
583 
482 

449 

15 
30 
13 
33 
14 
19 

16.5 
33-0 

14-  3 
36-3 

15-  4 
20.9 

1260 
2410 
^370 

2570 
1260 

I7922I  8.65 
34278  16.5 
19486  7.25 

?6t;c;A  26.8 

123 

235 
103 

385 

205 
232 

10.5 
10. 0 

9-3 
21.7 
10.3 
13-1 

149 
142 
132 
309 
147 
186 

7-4 
7.0 
6.0 

105-3 
99-6 
85-3 

17922 

14.4 

16.3 

7-6 

108. 1 

9-8 
14-5 

10.8 
16.0 

III5 
1580 

15859 
22473 

10.7 

15-0 

152 
213 

10.7 
9-9 

152 
141 

9-7 
9.0 

138.0 
128.0 

Breaking  Stage. 


Shearing  Stress 

(Si 

T  at  the 

B 

Load  Q  in 

Gb 

Ge 

Ti 

To 

Connection 
Between  the 

Slab  and 

B 

the  Rib. 

<D 

CQ 

B 

B 

B 

'c 

B 

B 

°B 

t 

0 

\ 

0 

\ 

\ 

0 

\ 

c 

_o 

M 

£ 

M 

M 

I 

25-7 

28.3 

38.0 

540 

2060 

29300 

13-9 

208 

16.9 

240 

10.4 

148 

II 

40.0 

44-0 

58.0 

825 

3150 

44803 

21  .  2 

302 

12.9 

183 

13-9 

208 

III 

19-5 

21-5 

28.0 

398 

i960 

27877 

10.3 

147 

13-2 

198 

8.1 

115 

VI 

42.0 

46.  2 

62.0 

882 

3260 

46368 

(33-5) 

476 

27.0 

384 

16. 7 

238 

V 

31.0 

34-1 

48-3 

687 

2600 

36981 

29-4 

418 

21  .0 

299 

13.0 

185 

VI 

37-8 

41 .6 

56.0 

797 

2950 

41959 

30.4 

432 

24-5 

348 

15-2 

216 

VII 

34.0 

37-4 

65-0 

925 

3420 

48643 

31.5 

448 

22.4 

319 

13.8 

196 

VIII 

23-4 

25-7 

47-6 

677 

2450 

34847 

22.8 

324 

22.8 

324 

14.7 

209 

IX 

25-6 

28.2 

52.2 

742 

2690 

38261 

24.8 

353 

17.7 

252 

II  .0 

156 

X 

27.0 

29.7 

(77-5) 

1102 

4050 
3880 

57605 
55187 

25-4 

361 

18. 1 

257 

II. 2 

159 

XI 

26.0 

28.6 

(83-0) 

1181 

4000 

56894 

158 

3800 

54049 

25.1 

357 

17.9 

25s 

II .  I 

XII 

26.0 

28.6 

(74-6) 

1061 

3900 

55471 

348 

3740 

53195 

24-5 

17-5 

249 

10.8 

154 

ACTION  OF  SHEARING  FORCES 


173 


of  the  concrete  would  occur  later,  in  spite  of  the  high  pressure  in  the  top  concrete 
layer. 

For  sake  of  clearness,  the  results  of  these  twelve  experiments  arc  reproduced 
in  Table  XXXI,  where  are  also  given  the  values  of  the  stresses  in  the  concrete 
at  the  first  crack,  based  on  Stage  I  with  ^  =  15,  and  the  shearing  stresses  in  the 
p^lane  connecting  the  stem  and  the  slab,  when  failure  took  place. 


CHAPTER  XI 


THEORY  OF  REINFORCED  CONCRETE 

STUTTGART  EXPERIMENTS  CONCERNING  SHEAR, 
CONTINUOUS  MEMBERS,  ETC.* 

These  experiments  carried  out  for  the  Eisenbetonkommission  der  Jubilaums- 
stiftung  der  Deutschen  Industrie,  were  performed  on  rectangular  and  T-beams 
and  should  give  information  concerning  the  value  of  ti  during  flexure.  In  the 
execution  of  the  programme  of  tests  it  must  be  stated  that  it  was  intended  that 
failure  of  a  specimen  was  not  to  take  place  in  any  other  manner  until  the  sliding 
resistance  of  the  reinforcement  had  been  exceeded. 

The  rectangular  beams  2.16  m.  (7.1  ft.)  long,  were  tested  on  a  clear  span  of 
2  m.  (6.56  ft.)  by  two  symmetrical  loads  i  m.  (3.28  ft.)  apart.  The  straight 
unhooked  rods  were  left  visible  at  the  ends  of  the  beams  to  that  the  smallest 
movement  could  be  measured,  and  over  the  supports  they  were  almost  entirely 
isolated  from  the  concrete  by  small  cavities  left  in  it.  The  failure  resulted  in  all 
cases  by  overcoming  the  frictional  resistance,  the  computed  values  of  which  are 
given  in  Table  XXXII. 


Table  XXXII 


No. 

Section. 

Reinforcement,  One  Rod. 

Values  at  Failure  Computed 
According  to  the  "Leitsatze." 

Age. 

cm. 

in. 

of  Diameter 

With 
Surface. 

TO 

^1 

mm. 

in. 

kg/cm  2 

Ibs/in2 

kg/cm2 

Ibs/in2 

I 

2 

3 
4 
5 

30/30 

30/30 
20/30 
15/30 
30/30 

II. 8/11. 8 

II.8/11.8 

7.9/ll.S 
5.9/11.8 
II.8/11.8 

25 

25 

18 
.  22 

32 

I 
I 

11 

T6 
7 
8 

li 

smoothed 

from  the 
rolls 
Do. 
Do. 
Do. 

/  2.7 
I3-8 

/4-7 
l5-7 
6.0 
8.8 
6.6 

38.4 
54-1 
66.9 
81. 1 

85-3 
125.  2 

93-8 

10.3 

14-5 
17.9 
22.0 
21 . 1 
19. 1 
19.8 

146.5 
206.  2 
254.6 
312.9 
300.1 
271.7 
281.6 

50  days* 
6  mos.f 
50  days| 
6  mos.J 

Do.§ 

Do.ll 

Do.^l 

*  0nly  one  vertical  crack  under  a  load.  §  Break  somewhat  inclined  in  part, 

t  Vertical  break.  ||  Break  clearly  inclined  in  part. 

t  Break  very  slightly  inclined.  ^  Break  almost  vertical. 


*  C.  V.  Bach.  Versuche  mit  Eisenbetonbalken,  Berlin,  1907.  Mitteilungen  iiber  Forschung- 
sarbeiten,  Nos.  45  to  47. 

174 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  175 


In  specimen  No.  i,  which  was  fifty  days  old,  the  failure  resulted  at  the 
appearance  of  the  first  crack,  while  in  all  other  cases,  the  frictional  resistance 
was  so  great  that  other  cracks  appeared  with  increase  of  load.  In  all  cases  a 
longitudinal  crack  formed  along  the  under  side  of  the  reinforcement  as  soon  as 
sliding  started,  growing  out  of  the  inclined  direction  of  the  cracks  as  above 
described.  Between  the  two  loads  where  the  moment  was  constant,  the  tension 
cracks  were  almost  uniformly  distributed.  The  carrying  power  of  this  beam 
ceased  with  the  slip  of  the  rod. 

Several  beams  similar  to  No.  4  were  built  with  stirrups  made  of  7  mm. 


U( 


approx.)    material,    which   clasped   the   reinforcing   rod   closely  and 


were  spaced  8  cm.  (3J  in.)  apart  between  the  supports  and  the  loads.  The 
center  portion,  between  the  loads,  where  no  lateral  forces  were  active,  had  no 
stirrups.    The  experiments  showed  that  the  first  tension  cracks  occurred  earlier 


Fig.  lOi. — Side  view,  bottom,  and  section  of  a  rectangular  beam  with 
Straight  reinforcement  and  with  stirrups. 


and  close  to  the  inner  stirrups,  which  is  to  be  ascribed  to  the 
weakening  of  the  concrete  section  at  those  points.  Moreover, 
the  direction  of  the  cracks  was  exactly  like  that  of  the 
beams  without  stirrups.  The  crack  causing  failure  which 
extended  diagonally  upward  to  the  point  of  application  of  one  load,  was,  however, 
more  inclined  (Fig.  161). 

The  average  value  of  the  computed  frictional  resistance  of  the  three  experi- 
ments was 


^1  =  23.3  kg/cm2  (331  lbs/in2), 


while  the  corresponding  beam  without  stirrups  gave  only  19. i  kg/cm^  (272  lbs/in^). 
This  increase  is  due  to  the  resistance  offered  by  the  stirrups  to  the  downward 
pressure  of  the  reinforcing  rods.  On  the  assumption  made  previously  in  connection 
with  diagonal  tension  cracks,  that  the  stirrups  resist  the  lateral  forces  over  a  distance 
along  the  neutral  axis,  equal  to  the  arm  of  the  couple  between  the  centroids  of 
tension  and  compression,  then  the  tensile  stress  in  the  stirrups  at  the  first  slip 
equaled  1680  kg/cm^  (23,895  lbs/in^).  The  stress  of  ti  =  23.3  kg/cm^  (331  lbs  in-) 
is  in  good  accord  with  the  results  of  direct  adhesion  experiments.  The  com- 
position of  the  concrete  was  similar,  viz.,  one  part  Portland  cement  to  four  parts 
Rhine  sand  and  gravel,  with  159^  of  water.    The  tensile  strength  of  the  con- 


176 


CONCRETE-STEEL  CONSTRUCTION 


Crete  was  ascertained  to  be  i2.6kg/cm2  (179  lbs/in^),  and  hence  the  inclination 
of  the  crack  causing  failure  would  be  slight,  with  the  very  small  value  of  Tq. 

ri  1  


Fig.  162. — Side  view,  bottom,  and  cross-section  of  a  beam  of  T-section 
with  straight  reinforcement  without  hooks  and  with  stirrups. 


The  beams  of  T-section  had  a  span  of  3  m.  (9.8  ft.), 
and  were  tested  with  two  symmetrically  placed  loads, 
I  m.  (3.3  ft.)  apart.  Of  the  three  groups,  all  of 
which  had  only  straight  main  reinforcing  rods,  two 
of  24  and  one  of  32  mm.  (Jf  and  ij  in.)  diameter, 
the  first  had  no  stirrups,  while  the  second  had  7  mm. 
in.  approx.)  stirrups  closely  clasping  the  rods,  and  spaced  9  cm.  (3.5  in.) 
apart  throughout  the  spaces  with  constant  shear,  and  the  third  had  thirty  by 


K  A  Fig.  163. — Side  view,  bottom,  and  section  of  a  beam  of  T-section  with 

straight  reinforcement  and  with  stirrups. 


2  mm.  by        in.)  flat  iron  individual  (Hennebique) 

stirrups  on  each  rod,  stirrups  14  cm.  (5.5  in.)  apart. 

In  the  group  without  stirrups,  the  first  cracks  appeared  at 
^^'^  loads  of  14  and  16  t.  (15.4  and  17.6  tons),  while  at  18  t. 
^^f^'^  (19.8  tons)  a  diagonal  crack  started,  running  toward  a  load 
point,  which  extended  very  well  up  the  side  and  soon  sur- 
passed the  center  tension  cracks  (Fig.  162).  The  further  course  and  final  appearance 
•at  failure  were  similar  to  those  of  Beams  I-III  of  the  T-beam  experiments 


K  -20-  >l 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  177 


■described  on  pages  152  to  159.  At  a  load  of  18  t.  (19.8  tons),  10  =  10.3  kg/cm^ 
(147  lbs/in^). 


Fig.  164. — Side  view,  bottom,  and  cross-section  of  beams  of  T-section 
with  straight  reinforcing-rods  and  stirrups. 


^  ^s. 


-  ^ 


The   second  group,   with  7  mm.  in.  approx.) 

stirrups,  developed  cracks  like  those  illustrated  in  Fig.  163, 
the  diagonal,  almost  straight  ones  at  an  angle  of  45°  ap- 
pearing last. 

In   the   beams   supplied   with   Hennebique  stirrups 
(Fig.    164),   of    the   cracks   between   the  load  and  a 
support,  the  lower  parts  followed  the  stirrups,  while  in  their  upper  parts  they 
took  an  inclined  direction  toward  the  load  points.    At  failure,  the  diagonal 


v. -20-^ 


Fig.  165. — Side  view,  bottom,  and  cross-section  of  "beams  of  T-section 
with  one  straight  and  four  bent  rods  without  stirrups. 

cracks  shown  in  the  illustration  near  the  support  were 
present. 

Other  beams  of  similar  dimensions  were  tested,  the  princi- 
pal reinforcement  of  which  consisted  of  one  round  rod  32  mm. 
(ij  in.)  in  diameter,  and  four  bent  rods  of  18  mm.  (f  in. 
approx.)  diameter.  The  slope  of  the  latter  was  some- 
what flatter  than  45°.  Three  specimens  were  without  stirrups 
7  mm.    (3^  in.  approx.)   closely  clasping  stirrups,  spaced  9  cm 


..^j-  


while  six  had 
(3i  in.)  in 


178 


CONCRETE-STEEL  CONSTRUCTION 


the  outer  thirds.  In  half  of  the  beams  the  lower  rod  was  absolutely  straight, 
while  in  the  other  half  a  right  angle  hook  was  provided.    The  directions  of  the 


 ^  Fig. 


1 66. — Side  view,  bottom,  and  cross-section  of  beams  of  T-section 
with  one  straight  and  four  bent  rods  and  with  stirrups. 


cracks  were  like  those  of  the  beams  with  only  straight  rods 
and  with  stirrups,  as  shown  in  Figs.  165-167. 

For  extra  clearness,  the  results  are  collected  in  Table 
XXXIV,  in  which  are  also  included  the  beams  of  2  m. 
(6.4  ft.)  span,  without  stirrups,  with  rods  bent  about  45° 
(Fig.  168). 

Concerning  the  table,  it  should  be  added:  That  in  the  beams  with  straight  tension 
reinforcement,  the  more  stirrups  were  provided,  the  later  was  the  occurrence  of  the 
I 


Fig.  167. — Side  view,  bottom,  cross-section,  and 
end  view  of  a  beam  of  T-section  with  a  striaght 
round  rod  32  mm.  (ij  in.  approx.)  in  diameter 
with  hook,  from  bent  round  rods  18  mm.  (y|-  in. 
approx.)  in  diameter  and  with  stirrups. 


first  slip  and  failure.  This  point  is  explained  by  the  condition  that  the  tensile 
strength  of  the  stirrups  prevented  the  downward  pressure  of  the  reinforcing  rods 
near  the  supports  after  the  appearance  of  diagonal  cracks.    The  more  th^ 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  179 


I  o 

i  hr 


X 
X 
X 

w 

pq 

< 


OC    o.  o 
o 


«  00 


I- 


in 

-r 

00 

o  <^ 

C) 


M  <N 


^      M  O 

On   O  -i- 

>0  00 


'i-  00  <r) 

ON    <N  On 


O  t-^ 
00  'l- 


CN 


M  6 


:S  C  C  ra  c 
^        oj        4/  o 


00  00  00 
O   O  On 


<U  CO 


•r 

o  o 
-a  Ci^ 

o  -5 
U  _ 


o  o  o 


vO  o  ^ 

r/-,    ro  >0  \0 
M    lo  o 
00 


O    O    O  O 
O 

M       O)       04  M 


00  in 
M    4  O 


CO  NO  CO  00  On  O 
M    LO    On  OO    1^  -1- 


NO     -t    -)-    '■'-J  On 
CO     -to  ro  00 

M     CN4     ro    04     ro  <^ 


M     (N     -f    01    CO  M 


M  NO  m  OO 

to  NO  O  00 

O  ^  M  O 

M  On  04 

CNi  r<o  CN|  -1" 


O  W  On  w  00 

00  lO  M  M  LO 

-t  On  -t  O    NO  On 

M  M  C4  (N      CN4  04 


>0  -i-  O  O  On  On 
m  O     ^<-^    w  On 

O     i-o   1^   m    O  '  M 


NO     LT-  O      On  O 


0  0  mo 


O    'O  r*^    O  lO 

<-0    O     »^  w  NO 


O  O  ^ 
CO    O  - 


II 

si 

O 

•£■5 


q  ci 


M3 


180 


CONCRETE-STEEL  CONSTRUCTION 


tension  in  the  stirrups  was  augmented,  the  harder  did  they  press  the  main 
rods  upward  against  the  concrete,  and  the  harder  did  the  latter  act  diagonally 
downward  from  above,  in  consequence  of  which  a  considerable  frictional  resist- 
ance was  developed,  so  that  failure  took  place  only  after  the  first  slip  had 
occurred. 

For  similar  reasons,  in  beams  supplied  with  both  bent  rods  and  stirrups,  a 
greater  frictional  resistance  was  developed  than  when  stirrups  were  absent.  The 
action  of  the  hooks  at  the  ends  of  the  straight  rods  resulted  in  an  increase  of  the 
load  from  41  to  46.5  t.  (45.1  to  51.2  tons).  In  Fig.  167  can  be  seen  the  result 
of  the  failure  of  the  concrete  because  of  the  straightening  of  the  hooks. 


Fig,  168. — Side  view  and  bottom  of  a  T-beam  of  2  m.  (6.56  ft.)  span  with  one  straight  round 
rod  32  mm,  (ij  in.  approx.)  and  bent  rods  18  mm.  in.  approx.)  in  diameter  without 
stirrups. 


If  the  tensile  stresses  are  computed  for  the  stirrups  of  the  beams  which  had 
only  straight  rods,  on  the  assumption  that  at  the  appearance  of  the  diagonal 
crack,  they  must  carry  the  whole  shear  over  a  length  equal  to  the  distance 
between  the  centroids  of  compression  and  tension,  the  values  found  in  Table 
XXXV  are  obtained. 


TABLE  XXXV 


Beam  Illustrated 
in  Figure. 

Stress  in  Stirrups 

at  the  First  Slip. 

at  Failure. 

kg/cm^ 

lbs/in2 

kg/cm2 

lbs/in2 

163 
164 

3450 
1340 

49271 
19059 

4200 
1750 

59738 
24891 

The  beam  shown  in  Fig.  167  had  its  end  so  constructed  that  failure  took  place 
inside  the  loads  by  compression  of  the  concrete  at  the  top.  In  the  remainder 
of  the  beams,  with  bent  rods,  it  may  be  supposed  that  as  soon  as  a  slip  took 
place  in  the  straight  rods,  the  nearest  bent  one  was  stressed  more  heavily,  so 
that  the  breaking  crack  was  formed  between  the  two  bent  bars.  The  nearest 
bent  bar  then  acted  as  the  tension  member  of  the  beam.    In  consequence  of  the 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  181 


great  stretch  of  the  reinforcement,  the  zone  of  compression  in  the  slab  would 
become  shallower  until  the  top  layer  of  concrete  crushed.    (See  Fig.  i66.) 

From  a  comparison  of  Figs.  162-167,  seen  that  the  diagonal  cracks 

produced  by  any  load,  extend  very  high  in  beams  with  only  straight  rods  and 
no  stirrups  and  for  which  tq  as  estimated  is  practically  equal  to  the  tensile  strength 
of  the  concrete.  They  occur  later  when  stirrups  or  bent  rods,  or  both  combined, 
are  present.    The  more  steel  is  cut  by  a  crack,  the  later  will  it  appear. 

DEDUCTIONS  FROM  THE  EXPERIMENTS 

So  far  as  the  relations  between  the  foregoing  experiments  go,  for  simply 
supported  T-beams,  the  following  deductions  can  be  drawn: 

1.  In  reinforced  concrete  beams,  near  the  supports  neither  a  pure  vertical 
shear  exists  nor  one  in  a  horizontal  direction,  but  rather  the  action  of  the  shearing 
forces  develop  inclined  cracks  in  the  vicinity  of  the  points  of  support.  At  these 
cracks,  the  tensile  strength  of  the  concrete  will  be  exceeded  by  the  diagonal 
principal  stress,  and  it  depends  upon  the  manner  of  loading,  breadth  of  span, 
and  arrangement  of  reinforcement,  whether  the  failure  will  take  place  in  the 
center  because  of  high  bending  moment,  or  near  the  supports  indirectly  through 
heavy  shearing  forces.  In  general,  the  diagonal  cracks  follow  the  directions  of 
the  stress  trajectories.  By  employing  stirrups  and  variously  arranged  bent  rods, 
the  direction  of  the  cracks  is  not  materially  altered,  but  the  inclined  cracks  near 
the  supports  occur  later,  showing  that  this  steel  diminishes  the  diagonal  tensile 
stresses  in  the  concrete.  The  strength  of  the  concrete  in  pure  shear  plays  no  part 
in  producing  security  against  indirect  failure  of  the  concrete  from  shear,  and, 
moreover,  the  horizontal  and  vertical  shearing  stresses  produced  during  the  bending 
of  a  reinforced  concrete  beam  are  to  be  considered  such  that  the  elemental  areas 
affected  by  them  are  not  perpendicular  to  the  direction  of  the  main  tension  and 
compression. 

2.  In  any  beam  in  which  a  failure  would  take  place  at  a  support  because  of 
lateral  forces,  when  only  straight  rods  are  employed,  the  supporting  power  will 
be  increased  through  an  arrangement  of  stirrups  and  bent  rods.  Their  use 
seems  particularly  advisable,  since  without  much  increase  of  material  a  greater 
load  is  assured. 

It  is  very  important  that  the  straight  rods  do  not  shp  at  the  supports,  since 
both  series  of  experiments  showed  a  very  favorable  action  by  the  hooks  at  the 
ends  in  the  increase  of  ultimate  load.  In  comparison  with  the  usual  right  angle 
or  blunt  hook  employed  heretofore,  the  arrangement  proposed  by  Considere, 
and  shown  in  Fig.  127,  is  of  great  value,  since  it  renders  unnecessary  the  com- 
putation of  the  adhesive  stress. 

From  the  tests  made  by  the  author,  it  is  shown  that  the  best  results  follow 
when  the  bending  of  rods  is  so  done  that  they  may  carry  the  diagonal  tensile 
stresses  equal  to  tq  which  act  at  an  angle  of  45°,  and  also  provide  the  necessary 
amount  of  steel  along  the  under  side  to  care  for  the  moments.  A  reinforced 
concrete  beam  of  constant  depth  can  then  be  compared  to  a  single  or  double 
intersection  truss  or  one  of  higher  order  (Figs.  143  and  144),  in  which  tension 


182 


CONCRETE-STEEL  CONSTRUCTION 


and  compression  members  slope  toward  the  middle  at  an  angle  of  45°.  From 
the  outset,  it  may  be  concluded  that  the  double  system  is  better  than  the  single, 
since  then  the  reinforcement  is  distributed  more  uniformly  through  the  concrete 
rib.  A  somewhat  flatter  slope  of  the  bent  rods  appears  of  no  value,  but  it 
may  be  recommended  for  constructive  reasons  in  large  spans  on  account  of 
conditions.  At  the  upper  ends  of  the  inclined  parts  of  bent  rods  the  force 
carried  by  the  tension  member  must  be  resolved  into  the  force  at  right  angles 
to  it  in  the  compression  member,  and  a  force  in  the  direction  of  the  top  chord. 
The  latter  component  acts  in  the  upper  part  of  the  bent  rod  itself,  and  it  must 
have  a  straight  portion  ending  with  an  effective  hook,  capable  of  transferring  its 
stress  to  that  of  the  concrete  in  the  too  chord.*    Similarly  the  tension  in  the 


Fig.  169. 


inclined  part  of  a  bent  rod  at  the  lower  bend  must  be  resolved  in  the  direction 
of  the  compression  diagonal  acting  at  that  point,  and  of  the  lower  chord,  clearly 
showing  that  the  simplest  and  most  effective  course  is  to  bend  upward  in  a  curve 
of  radius  r,  one  of  the  rods  rendered  unnecessary  by  the  reduction  in  the 
bending  moment.  However,  when  this  is  done  the  adhesion  on  the  remainder 
of  the  reinforcement  will  be  more  severely  taxed. 

As  is  shown  by  Beams  IV  and  VI  of  the  author's  experiments,  it  is  necessary 
to  keep  at  a  low  value  the  compression  of  the  concrete  at  the  bend  of  the  rod. 
If  the  pressure  acting  on  a  unit  area  of  the  projection  of  the  bend  be  computed, 
as  is  customary  with  rivets,  then  there  is  found  approximately,  if  d  represents 
the  diameter 

p  dr=S^Oe7Z — , 
so  that  ^ 

GeTt  d 

r  =  . 

Ap 

With  (T^  =  iooo  kg/cm^  (14,223  lbs/in^)  and  ^  =  6okg/cm2  (853  lbs/in^),  ^  =  136? 
approximately.  If  the  rods  are  bent  cold,  it  is  easier  to  do  so  with  an  even 
greater  radius.  It  is  recommended  that  the  reinforcing  rods  be  bent  up,  as 
soon  as  they  become  useless  because  of  reduction  of  the  moments  near  the  sup- 
ports, and  be  anchored  in  the  zone  of  compression,  as  described  above;  because, 
according  to  some  earlier  experiments  of  Wayss  and  Freytag  (see  second  edition 
of  this  book),  when  the  lower  rods  have  been  provided  simply  with  hooks,  cracks 
occur  very  early,  due  to  the  sudden  change  of  stress. 


*  Of  the  imaginary  truss. — Trans. 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  183 


3.  Stirrups  also  increase  the  carrying  power,  since  their  tensile  strength  resists 
a  failure  at  the  end  of  a  beam.  According  to  the  experiments  of  the  Stuttgart 
testing  laboratory,  they  increase  the  adhesion  of  straight  rods  in  this  same  manner. 
However,  if  the  principal  reinforcing  rods  are  arranged  as  above  described,  then 
the  stirrups  have  only  a  subordinate  statical  function,  and  can  be  considered  a 
further  item  of  security,  which  becomes  active  when  other  structural  elements  fail. 
Stirrups  are  also  useful  through  the  center  i)ortions  of  l)eams  when  the  latter 
are  unsymmetrically  loaded  and  shearing  stresses  are  produced  at  points  which 
are  not  usually  included  in  statical  computations.  Then  the  stirrups  must  act 
•as  vertical  tension  members,  as  illustrated  in  Fig.  141,  or  if  a  later  stage  be  considered, 
as  reinforcement  of  the  parts  of  the  concrete  rib  between  cracks.  Moreover,  an 
experiment  of  Schiile  (Table  VII,  No.  X,  Eidgenossischen  Materialpriifungsanstalt, 
Zurich)  with  a  heavily  reinforced,  uniformly  loaded,  T-shaped  beam  (No.  13), 
the  end  of  w^hich  was  reinforced  against  shearing  stresses  by  proper  bent  rods, 
showed  that  with  increase  of  load  the  characteristic  diagonal  cracks  which  finally 
produced  failure  appeared  in  the  central  portion  which  was  reinforced  against 
shear  neither  with  stirrups  or  bent  rods,  and  cut  directly  through  the  tension 
cracks  which  had  formed  earlier.  When  they  are  present,  stirrups  thus 
have  a  value  throughout  the  middle  of  a  beam  similar  to  the  one  they  possess 
at  the  ends. 

Evidently  it  is  not  wise  to  confine  the  stirrups  simply  to  the  ribs.  At  the 
same  time  they  assure  a  connection  between  the  rib  and  the  floor  slab  in 
cases  where  splitting  apart  of  the  concrete  might  occur.  Further,  it  is  believed 
that  a  beam  with  stirrups  throughout  its  whole  length  withstands  dynamic 
action  better  than  one  without  them.  The  computation  of  the  stirrups  can  be 
made  on  the  assumption  that  the  area  of  stirrups  cut  by  a  section  taken  at  an 
angle  of  45°  through  the  rib,  carries  all  the  lateral  forces  existing  in  that  sec- 
tion. In  the  centers  of  the  beams,  where  bent  rods  cannot  be  arranged,  the 
stirrups  must  be  designed  for  the  whole  shear,  while  near  the  supports  the  whole 
of  the  shearing  stresses  can  be  computed  as  carried  by  the  bent  rods,  or  a  part 
by  the  stirrups  also. 

When  it  is  intended  that  a  given  security  against  failure  be  provided  in  all 
parts  of  a  reinforced  concrete  beam,  it  is  not  proper  to  consider  the  distribu- 
tion of  stresses  under  a  safe  working  load  as  measuring  this  security.  Then 
it  is  the  resistance  just  before  failure  which  is  involved.  Consequently,  the 
diagonal  reinforcement  and  the  stirrups  should  not  be  computed  in  connection  with 
the  diagonal  tension  in  the  concrete,  since  they  would  already  be  overloaded 
at  the  moment  of  failure.  In  the  central  portions  of  beams  the  tensile  strength 
of  the  concrete  in  a  diagonal  direction  cannot  be  computed  as  active,  when 
cracks  have  already  been  produced  by  the  normal  tensile  stresses. 

If  the  tensile  strength  of  the  concrete  is  assumed  as  8  kg/cm^  (114  lbs/in^), 
then  precaution  against  shear  at  the  supports  is  unnecessary  if  tq  does  not  exceed 
2  kg/cm^  (28  lbs/in^).  This  condition  usually  exists  in  rectangular  sections,  like 
slabs,  but  nevertheless  it  is  usual  to  bend  upward  at  a  flat  angle  a  part  of  the 
reinforcement.  In  slabs,  stirrups  are  unnecessary,  since  failure  occurs  in  the 
center  under  usual  conditions,  invariably  Stage  I  still  being  present  near  the 
points  of  support  even  at  failure,  and  if  diagonal  cracks  should  occur  at  such 


184 


CONCRETE-STEEL  CONSTRUCTION 


points,  the  resistance  offered  by  the  concrete  to  the  downward  pressure  of  the 
reinforcement  is  much  greater  than  is  that  of  the  small  ribs  of  T-beams. 

Stirrups  placed  normal  to  the  lower  reinforcement  are  considered  the  most 
suitable.    Inclined  at  an  angle  of  45°,  they  would  seem  to  be  able  better  to 

carry  diagonal  tensile  stresses.  In  this 
position  the  stirrups  would  resist  ten- 
sion, but  there  is  difficulty  in  transferring 
their  stress  to  the  lower  reinforcement. 
They  tend  to  slip  along  the  rods  and 
push  off  the  concrete  cover  around  the 
rods  in  the  ribs,  as  was  observed  in 
some  early  experiments  made  by  Wayss  & 
Fig.  170.— Bursting  effect  of  loose  diagonal  Freytag  (Fig.  170).  A  solid  connection 
stirrups.  between  diagonal  stirrups  and  the  lower 

rods  is  troublesome  to  secure,  and 
hardly  practical.  An  erect  position  slightly  within  the  angle  of  friction  would 
be  somewhat  better  than  an  exactly  vertical  one. 

4.  The  necessity,  in  all  designs,  of  considering  the  adhesive  stresses  on  the 
lower  reinforcement  is  clearly  shown  in  the  foregoing  experiments.  In  all 
beams,  w^here  a  sHpping  of  the  straight  rods  was  observed,  failure  did  not 
immediately  result,  the  remaining  structural  parts  (stirrups  and  bent  rods)  often 
increasing  in  stress  until  stretched  beyond  their  capacity.  Especially  will  the 
nearest  bent  rod  assume  the  function  of  the  lower  chord,  in  which  case  in  a 
statical  sense  the  beam  becomes  one  of  variable  depth.  The  greater  is  the 
resistance  to  slipping,  the  greater  is  the  carrying  power.  (Compare  the  beams 
of  Fig.  147  with  148,  and  of  166  with  167). 

Thus  it  is  necessary  in  all  beams  which  act  as  if  of  constant  depth,  and  in 
which  all  elements  perform  their  desired  function  up  to  failure,  that  the  straight 
rods  should  also  possess  proper  security  against  slipping.  The  most  effica- 
cious element  appears  to  be  a  good  form  of  end  hook,  somewhat  like  Fig.  127 
and  as  additional  security  a  correspondingly  low  adhesive  stress  Ti  at  the  ends 
of  straight  rods.  Concerning  the  permissible  value  of  Ti,  opinion  has  been 
divided.  In  the  "  Leitsatze,"  a  value  of  7.5  kg/cm^  (107  lbs/in^)  is  suggested, 
while  the  Prussian  Regulations  allow  only  4.5  kg/cm^  (64  lbs/in^).  In  both  cases, 
the  formula, 

  bzQ  

circumference  of  reinforcement ' 

is  given,  but  in  the  corresponding  example  in  the  "  Leitsatze,"  for  the  circum- 
ference of  the  reinforcement  only  that  of  the  straight  rods  is  considered,  while 
according  to  the  old  Prussian  regulations,  all  the  steel  can  be  figured.  This 
fact  was  not  considered  by  those  who  accuse  the  "  Leitsatze  "  and  the  author 
of  poor  judgment. 

The  point  especially  involved  in  the  question  concerning  on  which  reinforce- 
ment the  adhesion  acts,  follows  immediately  from  the  experiments,  since  if  only 
the  straight  rods  are  active,  a  slip  would  be  observed  in  them,  producing  a 
gradual  redistribution  of  internal  forces  which  would  lead  to  failure.  Rods 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  185 


which  are  bent  at  three  points  cannot  slip.  Stretching  and  local  shifting  of  the 
diagonal  rods  with  respect  to  the  concrete  will  certainly  take  j)lace,  but  when 
they  are  anchored  within  the  zone  of  compression,  as  shown  in  Fig.  169,  any 
real  slip  will  be  prevented.  If  the  hypothetical  value  of  ri  is  sought  with  respect 
to  the  circumference  of  only  the  straight  rods,  based  on  the  hypothesis  of  as 
perfect  a  supporting  power  as  a  beam  having  only  bent  rods — then  ti  must  ec^ual 
00.  Naturally,  this  is  impossible,  and  as  soon  as  such  a  beam  is  no  longer  exactly 
straight,  it  should  be  computed  as  possessing  a  bent  tension  chord. 

According  to  the  experiments  on  rectangular  beams  with  straight  rods,  and 

on  the  basis  of  the  formula  ri=^,  an  excellent  agreement  is  found  with  the 

values  of  direct  adhesion  experiments,  but  it  must  be  noted  that  this  formula 
is  not  strictly  applicable  when  bent  rods  are  present.    In  this  case,  two  ways 

are  open — either  the  formula  ti=^^  is  simply  assumed  and  the  correspond- 
ing values  ascertained  from  the  experiments  (which,  obviously,  will  not  agree 
with  those  of  direct  adhesion  experiments  but  will  represent  simply  comparative 
values),  just  as  was  done  in  regard  to  the  Navier  bending  formula  to  find  the 
bending  strength  of  concrete;  or  it  may  be  assumed  that  in  bending,  the  same 
adhesion  will  be  developed  as  in  direct  experiments,  and  endeavor  is  then  to  be 
made  to  find  a  suitable  formula  for  ri.  Both  courses  lead  to  the  same  result,  as 
far  as  practical  design  and  security  are  considered,  since  that  value  is  used  in 
design  which  has  just  been  derived  by  the  same  method  from  experiments. 
(See  also  page  97). 

In  Table  XXXVI,  the  values  found  experimentally  of  ri  at  the  first  slip,  are 
again  collected. 

Table  XXXVI 


Beam. 


bn~a 


kg/cm^i  lbs/in  2 


2U 


kg/cm^  lbs/in^ 


Ti  Considering 
all  Rods 


kg/cm2  lbs/in2  Rods 


Variety  of  Reinforcement 


Ends  Stirrups 


Wayss  and 

Frey  tag  „ 

Stuttgart,  Table 
No.  XXXIII, 

Beam  No  


Stuttgart,  Fig. 
No  ^ 


jlV 


3 
4 
5 

161 
162 
163 
164 

165 
166 
167 
168 


53-6 
32.6 
22.0 
21 . 1 
10. 1 
19.8 

23-3 
II .  I 
12.8 
•7 
•7 
.0 

■  4 

■7 


762 
474 
313 
300 
272 
282 

331 
158 
182 
20Q 
408 
498 
546 
522 


26.8 
16-3 


14.4 

17-5 
19.2 

18.4 


381 
237 


205 
249 

273 
262 


15.3 
9-3 
22.0 
21 . 1 
19. 1 
19.8 

23-3 
II . I 
12.8 
14.7 
8-7 
10.6 
II. 6 
II .  I 


218 
132 

303 
300 

272 
282 

331 
158 
182 
209 
124 
151 
165 
158 


Bent 
Bent 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Bent 
Bent 
Bent 
Bent 


Hooked 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Straight 
Hooked 
Straight 


For  the  beams  with  bent  rods,  the  value  of  ti  in  the  first  column  agrees 
fairly  with  that  of  the  experiments  in  which  the  rods  were  pushed  through  at 


186 


CONCRETE-STEEL  CONSTRUCTION 


high  speed.    Since  in  this  former  case  the  shpping  takes  place  slowly,  the  computed 
values  are  too  high.    A  better  agreement  is  observed  in  the  value  computed  by 
Q 

the  formula  ri=  in  the  next  column,  on   the   assumption   of   a  trussed 

2ZU 

condition  of  the  beam,  when  compared  with  the  value  found  from  rectangular 
beams,  and  also  from  direct  experiments  at  slow^  speed.  T-beams  without 
ibent  rods  gave  somewhat  lower  results,  since  the  downward  pressure  of  the  rods 
•was  less. 

According  to  experiments  of  the  Stuttgart  Testing  Laboratory*  the  sliding 
resistance  of  embedded  lengths  of  lo  to  30  cm.  (4  to  12  in.  approx.)  had  an 
average  of  15.3  to  25.1  kg/cm^  (218  to  357  lbs/in^). 

From  the  third  column  it  is  seen  how  inconsistent  it  is  to  consider  the  circum- 
ference of  the  bent  rods,  in  computing  zi.  Since  the  beams  with  bent  reinforce- 
ment developed  less  adhesion  than  the  beams  containing  only  straight  rods, 
the  conclusion  could  be  drawn  that  a  lessening  of  the  adhesion  took  place  with 
bending,  which  cannot  be  the  case.  It  is  readily  seen  that  with  a  permissible 
adhesive  stress  of  4.5  kg/cm^  (64  lbs/in^),  and  taking  into  account  all  rods, 
under  these  circumstances  a  factor  of  safety  of  only  about  two  is  secured  against 
slip  of  the  straight  rods. 

According  to  the  experiments,  the  permissible  adhesive  stress  for  slabs  or 
beams  without  bent  reinforcement,  on  the  basis  of  a  safety  factor  against  slip 
of  four,  can  be  taken  at  about  5  kg/cm^  (71  lbs/in^),  when  failure  is  not  to  be 
feared  from  the  shearing  stress  tq.  If  that  stress  is  considered  too  high,  a  valua 
of  3.5  kg/cm^  (50  lbs/in^)  should  suffice.  The  security  against  diagonal  tensile 
cracks  and  their  damaging  results  is  not  increased  by  this  means,  however. 

If  bent  rods  are  employed  in  such  amount  that  they  can  carry  all  the  diagonal 
tensile  stresses,  then  a  factor  of  safety  of  four  can  be  secured  with 


or 


7.5  kg/cm^  (107  lbs/in^)  in  connection  with  the  formula  tj  =-^> 


3.75  kg/cm^  (53  lbs/in^)  in  connection  with  the  formula  zi=^^~; 

2  (J 


in  which  U  represents  the  circumference  of  the  straight  rods  which  extend  over 
the  supports;  and  the  hooked  ends,  which  should  always  be  used,  will  increase 
the  security  factor  so  that  it  is  more  than  five. 

If  the  bent  rods  are  not  all  so  arranged  as  to  resist  the  diagonal  tensile 
stresses  tq,  then  a  part  of  the  lateral  forces  can  be  carried  by  stirrups.  For 
this  part,  ri  is  to  be  computed  according  to  the  first  formula,  while  for  the 
rest,  the  second  formula  is  to  be  used,  and  a  stress  chosen  between  3.75  and 
5  kg/cm^  (53  and  71  lbs/in^),  according  to  the  amount  of  Q  taken  by  the  bent  rods 
and  the  stirrups. 

*  Versuche  iiber  den  Gleitwiderstand  einbetonierter  Eisen,  by  Bach,  Berlin,  1905,  or  also 
No.  22  of  the  Mitteilungen  iiber  Forschungsarbeiten  auf  dem  Gebiete  des  Ingenieurwesens. 


SHEAR  AND  CONTINUOUS  MEMBER  EXPERIMENTS  187 


It  is  recommended  in  this  connection  that  the  whole  of  the  diagonal  tension 
near  the  points  of  support  be  taken  by  the  bent  rods,  even  though  stirrups 
be  employed  throughout  the  whole  length  of  the  beam,  and  effective  hooks  are 
placed  on  the  ends  of  the  straight  rods. 

5.  With  reference  to  the  foregoing  recommendation,  the  method  can  be 
followed  which  is  shown  in  the  sketches  of  T-beams  of  Figs.  171  and  172. 

In  Fig.  171  a  double  intersection  truss  is  assumed,  in  which  the  bent  rods 
Si,  S2,  and  S3  are  designed  to  carry  the  whole  of  the  diagonal  tension  tq.  Accord- 
ing to  the  method  of  Fig.  142  and  page  160,  the  forces  ^i,  S2,  and  Ss  are  repre- 
sented by  the  shaded  areas  shown  at  an  angle  of  45°,  which  areas  are  to  be  multi- 


])lied  by  the  breadth  ^0  of  the  rib. 
The  bending  of  the  rods  will  reduce 
the  resisting  moment  of  the  beam 
toward  the  ends,  but  the  line  of 
maximum  moments  can  be  used 
to  ascertain  whether  a  sufficient 
carrying  power  is  present.  It  is 
thus  easy  to  compare  with  the 
area  representing  the  maximum 
moments  the  area  showing  the 
moment  which  the  beam  can  carry  at  each  point,  so  that  the  lower  reinforce- 
ment may  not  be  stressed  beyond  the  allowable  limit  at  the  points  of  bend. 
Where  the  bent  rods  are  discarded  for  resisting  the  moment,  the  method  gives 
results  favorable  to  the  moment  diagram  and  therefore  on  the  safe  side. 

The  reentrant  angles  of  the  corresponding  polygon  should  lie  outside  the 
curve  bounding  the  area  of  maximum  moments.  So  much  straight  reinforce- 
ment must  be  carried  over  the  supports  that  zi  will  be  between  7.5  and  3.75  kg/cm^ 
(107  and  54  lbs/in^). 

When  the  bending  is  done  so  as  to  correspond  with  the  diagonal  members  of 
a  double  intersection  truss,  the  forces  Si,  S2,  and  Ss  can  also  be  determined  by 
a  resolution  of  the  lateral  forces  of  the  corresponding  panels,  as  is  illustrated  in 
the  lower  diagram  of  Fig.  171.     In  a  single  intersection  system  S  would  equal 


188 


CONCRETE-STEEL  CONSTRUCTION 


If  computations  are  made  with  partial  live  load,  as  it  is  well  to  do  in  all  cases, 
then  the  lateral  forces  are  not  zero  at  the  center,  and  to  care  for  them  the  bend- 
ing of  the  rods  must  be  started  so  soon  that  not  enough  steel  will  remain  to  care 
for  the  moments.  Although  a  certain  arch-like  dimunition  of  shearing  stress  takes 
place,  it  seems  wise  to  provide  some  structural  elements.  As  such,  stirrups 
are  most  easily  available,  and  they  can  be  computed  on  the  assumption  that 
they  carry  all  the  lateral  forces  in  a  length  equal  to  z.  If  e  is  the  stirrup  spacing, 
then  the  force  in  a  single  stirrup  is 


z 


For  general  reasons  already  given,  a  certain  stirrup  spacing  (20  to  30  cm. — 
8  to  12  ins.)  should  not  be  exceeded,  and  they  should  be  employed  even  where 
the  rods  are  bent.  As  a  further  precautionary  measure  the  inner  bend  may  be 
somewhat  flatter,  or  a  greater  rounding  of  its  angles  be  made,  as  is  shown  by 
the  dotted  line  on  the  right-hand  half  of  the  sketch. 

In  Fig.  172  the  bent  parts  are  closer  together,  and  a  part  of  the  lateral  force 


the  area  for  tq.    If  the  tensile  strength  of 

a  single  stirrup  is  B,  then  the  breadth  of  this  strip  is  — .    The  bent  rods  cor- 

eho 

respond  to  equal  parts  of  the  ro-area,  if  the  rods  are  all  of  the  same  size. 

The  German  "  Ausschuss  flir  Eisenbeton"  also  includes  in  its  program,  experi- 
ments to  make  clear  the  action  of  shearing  forces.  Upon  completion  of  these 
tests,  opportunity  will  be  given  of  proving  the  accuracy  of  the  ideas  and  methods 
here  set  forth. 

6.  Security  against  cracking  of  T-beams.  While  it  can  be  concluded,  from 
bending  tests  of  rectangular  reinforced  concrete  beams,  that  sufficient  security 
against  the  appearance  of  the  first  tension  crack  is  provided  when  the  methods 
of  design  contained  in  the  "  Leitsatze  "  are  followed,  as  a  matter  of  fact  the 
same  has  not  yet  been  etablished  for  beams  of  T-section.    Here,  the  amount  of 


SHEAR  AND  CONTINUOUS  MEMBER  EXPETRIMENTS  189 


reinforcement  is  much  greater  in  relation  to  the  concrete  subjected  to  tension,  so 
that  the  tensile  stresses  in  the  concrete  are  not  great  enough  to  diminish  those 
in  the  reinforcement  so  that  tension  cracks  in  the  concrete  will  be  avoided.  Con- 
scquendy  the  security  against  cracking  decreases  with  increase  of  the  percentage 
of  reinforcement  in  the  ribs.  According  to  the  experiments  cited  in  Table  XXXII, 
page  172,  in  Beams  I,  III,  IV,  V,  VI,  with  a  reinforcement  amounting  to  2.18% 
of  the  area  of  the  rib,  the  first  crack  was  observed  at  a  stress  computed  at  approx- 
imately 700  kg/cm^  (9956  lbs/in^  (according  to  Stage  II  b).  In  Beam  II,  with 
1.09%  of  reinforcement,  the  computed  steel  stress  rose  to  1200  kg/cm^  (17068 
lbs/in^).  The  tension  in  the  concrete,  computed  according  to  Stage  I,  with  ^  =  15, 
is  given  in  the  several  descriptions  of  the  tests,  and  varied  between  20  and  27 
kg/cm^  (284  and  384  lbs/in^).  The  results  of  the  experiments  of  the  Stuttgart 
Testing  Laboratory  concerning  the  appearance  of  the  first  crack,  are  gathered 
in  Table  XXXVII.  The  amount  of  reinforcement  in  the  T-sections  is  given 
as  a  percentage  of  the  concrete  area  between  the  bottom  of  the  beam  and  the 
top  of  the  slab  and  of  a  breadth  equal  to  that  of  the  rib. 


Table  XXXVII 


Beam 

Reinforce- 
ment 

% 

First  Crack  Observce 

between  the  Limits. 

Steel  Stresses  According  to 
Stage  II  6 

Concrete  Tensile  Stresses  Accord- 
ing to  Stage  I  with  m  =  15 

Figure  No. 

Stirrups 

kg/cm  2 

lbs/in2 

kg/cm^ 

lbs/in  2 

162 

None 

1.79 

773-839 

I 0994-1 1947 

33-3-36-0 

474-512 

163 

Used 

1.79 

747-812 

10624-11549 

52.3-35-^ 

459-501 

164 

Used 

1.79 

642-728 

9131-10354 

27-7-31-4 

394-447 

165 

None 

1.82 

725-808 

10312  -11492 

32.3-36.0 

459-512 

166 

Used 

1.82 

680-744 

9672 -10582 

29.6-32.3 

421-459 

167 

Used 

1.82 

699-762 

9942-10838 

30.5-33-3 

434-474 

168 

Used 

1.82 

753-785 

I1477-11165 

33-2-34-6 

472-492 

The  directly  observed  tensile  strength  of  the  concrete  was  about  13  kg/cm^ 
(185  lbs/in2). 

Similar  stresses  in  the  reinforcement,  between  600  and  700  kg/cm^  (8534 
and  9956  lbs/in^)  w^ere  observed  by  Schiile,  and  reported  in  the  Mitteilungen 
der  Eidgenossischen  Materialpriifungsantalt,  Zurich,  No.  X. 

The  first  tension  cracks  in  the  concrete,  which  are  so  fine  that  they  cannot 
be  discovered  on  rough,  unpainted  surfaces  of  beams,  need  give  no  anxiety 
unless  the  tensile  strength  of  the  concrete  has  been  included  in  making  calcula- 
tions. This  should  rarely  be  done,  however.  When,  with  the  usual  arrangement, 
the  practically  invisible  cracks  are  exactly  crossed  by  a  needful  amount  of  reinforce- 
ment, there  is  nothing  to  fear,  because  this  condition  is  found  in  the  greater 
number  of  well  constructed  reinforced  concrete  structures,  many  of  which  are 
subjected  to  very  severe  conditions.  No  danger  of  rust  need  be  considered,  since 
the  covering  which  affords  protection  against  it  does  not  consist  of  the  porous 
concrete  but  rather  of  the  cement  film  immediately  covering  the  steel.  Further- 
more, the  first  cracks  along  the  edges  do  not  usually  extend  entirely  to  the  reinforc- 
ing rods. 


190 


CONCRETE-STEEL  CONSTRUCTION 


Nevertheless,  if  it  is  desired  to  design  T-beams  which  will  be  wholly  free 
from  cracks,  it  is  necessary  to  use  broader  concrete  ribs  or  less  reinforcement. 
In  consequence,  however,  the  design  of  many  buildings  with  T-beams  will  be 
uneconomical,  and  it  will  be  better  to  employ  some  form  of  arch  construction. 
In  usual  building  work,  absolute  freedom  from  rusting  is  generally  unimportant, 
so  that  excessive  care  need  not  be  exercised. 


In  the  use  of  reinforced  concrete,  it  often  happens  that  the  depth  of  a  beam 
is  increased  where  the  moment  is  greatest.  Figs.  173-176  show  the  usual  arrange- 
ments for  positive  and  negative  bending  moments. 

The  direction  of  the  section  to  be  made  for  purposes  of  computation  cannot 
be  assumed  at  random,  since,  in  the  neighborhood  of  the  outside  layers,  the 
compressive  stresses  act  parallel  with  them,  and  the  steel  stresses  naturally  act 
in  the  direction  of  the  rods.  In  order  to  simplify  the  derivation  of  formulas  for 
To,  however,  the  section  will  be  assumed  as  vertical,  since  otherwise  it  would 
involve  lateral  forces,  and  should  also  rigorously  take  acount  of  bending  and 
of  axial  pressure.  For  all  the  cases  shown  in  Figs.  173-176,  when  only  vertical 
loads  are  assumed 


so  that  the  increment  of  Z  in  the  adjacent  section  is 


SHEARING  STRESSES  IN  BEAMS  OF  VARIABLE  DEPTH 


dZ 


zdM—Mdz 


or 


dZ, 
dl 


ldM_M  dz 
z   dl     z^  dl' 


Further,  in  all  cases, 


and 


h  ToXdl  =  U  TiXdl=dZ. 


In  Figs.  175  and  176, 


_dl___dZ_ 
cos  a    cos  a 


The  arm  2  ot  the  couple  between  tension  and  compression  may  be  assumed  as 
equal  to  J  h  for  all  practical  purposes,  so  that,  with 


SHEARING  STRESSES 


191 


there  is  obtained 


Q  7 

b  To  =  U  Ti=  — tan  a. 

Z      5  2" 


This  is  a  less  value  than  if  the  height  were  constant,  since  the  second  quantity 

is  subtracted  from  — .    If  the  formula  is  written 
z 


bTo  = 


Q-|-  — tana    Q-^-Z  tan  a  Q-^Dinna 
8   2  8  8 


it  is  evident  that  in  comparison  with  a  beam  of  constant  depth,  instead  of  the 
total  shear,  a  value  found  by  diminishing  it  by  |  D  tan  a  is  to  be  used  in  com- 
puting To. 

If,  in  Figs.  173-174,  the  resultant  pressure  is  assumed  as  inclined  in  the  direc- 
tion of  a  line  connecting  the  centroids  of  compression  in  adjacent  sections,  then 


z  A 


Fig.  173. — Positive  Bending  Moment. 


Fig.  174. — Negative  Bending  Moment. 


 '  X  

i  ^ 

D 

Z 

Fig.  175. — Positive  Bending  Moment. 


Fig.  176. — Negative  Bending  Moment. 


since  D  must  be  its  horizontal  component,  the  subtractive  quantity  J  D  tan  o: 
can  be  none  other  than  its  vertical  component.  Since  the  compressive  stresses 
act  on  layers  parallel  to  this  direction,  this  deduction  is  entirely  plausible.  In 
Figs.  175-176,  the  resultant  pressure  may  be  assumed  as  acting  in  the  direction 
of  the  line  connecting  its  points  of  application,  and  the  quantity  represents  the 
difference  between  the  vertical  components  of  the  upwardly  directed  pressure 
and  the  downwardly  inclined  tension. 

In  the  equation  for  h  tq  is  seen  the  beneficial  influence  of  arching  the  inter- 


192 


CONCRETE-STEEL  CONSTRUCTION 


mediate  sections  of  a  continuous  reinforced  concrete  beam.  If  the  moment  in- 
creases with  decrease  of  //,  then  the  minus  sign  in  the  formula  is  to  be  changed 
to  plus. 

DEFORMATION 

Concerning  the  experiments  already  described  on  page  105,  it  was  said  that  the 
concrete,  because  of  its  tensile  strength,  relieved  the  stress  in  the  reinforcement 
to  some  extent,  and  that  in  Stage  11^  this  condition  also  existed  to  an  even  greater 
extent.  Because  of  this  action,  the  deflections  of  reinforced  concrete  struc- 
tures are  generally  very  small.  It  is  to  be  further  noted  that  because  of  the  fixed 
connection  between  all  parts  of  a  reinforced  concrete  structure,  more  structural 
parts  contribute  to  the  support  of  the  loads  than  are  usually  considered  in  making 
computations. 

In  researches  concerning  structural  bridges,  deformations  are  given  a  con- 
siderable importance  at  the  present  time,  but  without  proper  foundation  it 
would  seem,  since  the  total  deformation  is  the  result  of  a  large  number  of  very 
small  elastic  deformations  of  the  various  parts  and  sections.  Thus,  it  is  impossible 
to  trace  in  the  computed  total  deformation  the  effect  of  one  or  more  defects  in  a 
member,  such  as  a  poor  rivet,  etc.,  comprising,  as  it  would,  such  a  small  part  of 
the  whole;  but  a  fairly  exact  determination  can  and  should  be  made  of  the  whole 
structure,  on  the  basis  of  known  causes. 

The  amount  of  deformation  of  any  reinforced  concrete  construction  is  of 
even  less  value  as  a  measure  of  its  quality,  since  adequate  determinations  of  the 
influence  of  shearing  and  adhesive  stresses  are  wanting,  and  since  the  distribu- 
tion of  load  within  the  construction  cannot  be  followed  with  mathematical 
exactness.  If  one  cannot  leave  matters  to  the  experience  of  the  company  execut- 
ing the  work,  nothing  remains  but  to  become  familiar  with  the  details  of  construc- 
tion and  design  in  order  to  be  sure  of  the  necessary  care  during  construction. 

When  exact  experimental  knowledge  as  to  the  safety  of  the  structure  is  to  be 
brought  into  the  question,  the  size  of  the  actual  stresses  produced  in  steel  and  concrete 
should  be  known,  and  they  can  best  be  determined  during  an  experiment  by 
means  of  a  suitable  measuring  device  (such  as  a  Rabut-Manet).  In  this  in- 
stance the  deformations  are  not  determined  indirectly. 

In  Fig.  177  are  given  the  deflection  diagrams  of  the  tests  described  on  page 
95  (Fig.  88).  The  constant  bending  moments  at  the  centers  are  plotted  as 
abscissas,  and  the  resulting  deformations  as  ordinates.  The  circles  on  each 
curve  show  the  permissible  loading  according  to  the  "  Leitsiitze." 

The  condition  of  the  curves  up  to  these  points  is  practically  rectilinear,  but 
at  the  appearance  of  the  first  crack  an  upward  turn  takes  place.  That  is,  a 
sudden  increase  in  the  deflection  occurs,  so  that  the  conclusion  may  be  drawn 
that  in  reinforced  concrete  slabs,  cracks  actually  exist  at  those  points.  For 
T-beams,  the  courses  of  the  curves  are  similar. 

According  to  these  diagrams,  the  action  of  a  flexed  reinforced  concrete  beam 
is  such  as  to  give  a  deflection  diagram  consisting  of  two  straight  lines  with  a 
transition  curve  between  them.  The  first  part  starting  from  the  origin  cor- 
responds with  stresses  in  Stage  I,  in  which  the  concrete  yet  exerts  tension.  The 


DEFLECTION 


193 


broken  line  connecting  the  two  parts  corresponds  with  Stage  Ua,  in  which 
the  tensile  strength  of  the  concrete  has  reached  its  ultimate  point,  and  finally 
the  commencement  of  Stage  11^,  where  at  several  points  the  tensile  strength 
has  been  exceeded  and  fine  cracks  appear.  The  further  practically  recti- 
linear character  of  the  curve  corresponds  with  Stage  lib,  with  increasing  cracks, 
during  which  the  reinforcement  is  prevented  from  stretching  indefinitely  only 
by  the  practically  constant  condition  of  the  resistance  offered  to  its  sliding,  by 
its  concrete  covering.  Consequently,  here,  the  deflection  is  not  proportional  to 
the  load. 


Fig.  177. — Deflection  diagrams  of  beams  of  pp,  100,  loi. 

A  mathematical  determination  of  the  deflection  must  thus  fit  these  several 
conditions.  For  Stage  I,  computations  can  be  carried  out  according  to  the  usual 
theories  of  flexure,  but  the  expression  for  the  moment  of  inertia  must  have  the 
sections  of  reinforcement  replaced  by  w-fold  larger  ones  of  concrete.  The  kind 
of  deflection  in  Stage  11^  makes  its  computation  impossible,  since  its  determina- 
tion rests  on  the  average  stretch  in  the  steel,  the  spacing  of  the  tension  cracks 
and  the  frictional  resistance  between  the  steel  and  concrete.  Also,  a  method  of 
computation  for  Stage  lla,  during  which  the  stress  condition  (up  to  the  appear- 
ance of  the  first  crack)  is  quite  accurately  known,  is  equally  shown  to  be  out  of 
the  question,  since  the  special  formulas  (p.  138)  for  :v  and  rr;,  show  that  it  is  not 
possible  to  find  a  serviceable  expression  for  the  angle  between  the  adjacent 
sections,  when  it  is  recognized  that  /?  must  first  be  approximately  determined 
through  deductions  from  experiments. 


194 


CONCRETP]-STEEL  CONSTRUCTION 


COMPUTATION  OF  FORCES  AND  MOMENTS 

In  the  foregoing  pages  it  has  been  shown  how  to  compute  the  stresses  in  a 
section,  produced  by  known  moments  and  normal  forces,  and  how  structural  parts 
should  be  designed.  In  the  following  paragraphs  will  be  discussed  the  general 
case  of  the  computation  of  lateral  forces  and  moments. 

While  it  is  usually  sufficient  to  know  simply  the  maximum  moment  which 
may  occur  at  any  point  within  the  span  of  a  steel  beam  of  constant  area  (such  as 
any  rolled  section),  and  in  some  cases  also  of  plate  girders;  for  an  economical 
design  of  reinforced  concrete  beams,  the  inaximum  moments  of  both  kinds  must 
be  known  for  a  large  number  of  sections.  Above  all,  it  is  necessary  to  know  in 
which  direction  the  moment  acts,  since  the  location  of  the  reinforcement  in  the 
beam  depends  upon  it.  As  has  been  shown,  the  lateral  internal  forces,  such  as 
the  shearing  stresses,  play  a  considerable  part  in  the  design  of  the  sections  of 
reinforced  concrete  beams.  A  further  condition  which  must  be  included  in 
designing  is  that  the  permissible  stresses  of  each  of  two  different  materials  must 
not  be  exceeded.  On  the  other  hand,  a  much  simpler  form  of  cross-section  is 
to  be  dealt  with. 

For  all  statically  determined  reinforced  concrete  construction,  the  bending 
moments  are  to  be  computed  from  the  exterior  forces  according  to  the  rules  of 
statics.  The  question  arises,  however,  whether  for  statically  indeterminate 
reinforced  concrete  construction,  such  as  restrained  and  continuous  beams, 
arches  without  hinges,  etc.,  the  stresses  are  to  be  determined  in  the  same  manner 
as  for  homogeneous  materials. 

It  was  shown  by  Spitzer  in  the  Zeitschrift  des  Osterreichischen  Ingenieur- 
und  Architektenvereins,  in  connection  with  the  Purkersdorfer  test  of  a  Monier 
arch,  that  his  method  of  calculation  may  be  followed,  as  may  also  the  elastic 
theory  as  applied  to  homogeneous  materials,  when  the  area  of  reinforcement  is 
replaced  by  an  w-fold  greater  area  of  concrete  in  all  expressions  for  areas  F  and 
moments  of  inertia  /.  It  is  to  be  noted  that  in  bending  with  axial  pressure,  as  in 
an  arch,  the  tensile  strength  of  most  sections  is  not  involved,  since  the  action  of 
the  moment  and  of  the  pressure  are  to  be  added  in  the  same  manner  as  for  homo- 
geneous sections. 

A  restrained  or  continuous  T-beam  in  which  no  axial  force  acts  will  not  be 
considered.  So  long  as  the  angle  of  inclination  between  two  adjacent  sections  is 
proportional  to  the  bending  moment,  the  methods  of  computation  for  restrained 
or  continuous  beams  are  to  be  followed.  According  to  the  deflection  diagram, 
this  proportionality  between  moment  and  deformation  in  rectangular  sections 
continues  up  to  the  maximum  permissible  loading.  But  even  when  the  propor- 
tionality ceases  with  higher  loading,  the  stress  distribution  is  not  greatly  altered, 
as  the  following  simple  cases  show: 

As  an  extreme  case  it  will  be  assumed  that  the  angle  of  inclination  between 
the  adjacent  sections  is  proportional  to  the  third  power  of  the  moments  (so  that 
the  deflection  diagram  of  Fig.  177  would  be  a  cubical  parabola),  and  thus 

d(j)=^CMHx, 

wherein  C  represents  some  constant. 


THEORIES  OF  MOMENTS 


195 


For  a  beam  fixed  at  both  ends,  and  carrying  a  concentrated  load  P  at  the  center, 


2 


It  must  happen  that 


The  solution  gives  the  well-known  moment 


8  • 


■M, 


P. 


 /  - 

Fig.  178. 


An  unsymmetrical  case  will  be  selected  of  a  beam  in  which  one  end  is  fixed 
and  the  other  freely  supported,  Fig.  179, 

Mx^A  x-^x^, 


A  ^—j      )  dx. 


Since  the  end  of  the  beam  at  A  cannot  move  vertically, 

0=  )xd(l), 


or 


0=j'^^x(^Ax- 


x^  I  dXf 


The  equation  derived  from  the  integration  of  this  expression  is  satisfied  with 

A  =  ^gl,  the  known  proper  value  on  the  basis  of  a  frictionless  support. 

0 

It  may  be  concluded  from  these  two  cases  that  a  relation  differing  from 
true  proportionality  between  moment  and  deformation  makes  no  appreciable 
change  in  the  section,  from  that  found  by  pure  elasticity  formulas.  The  propriety 
of  computing  continuous  reinforced  concrete  beams  as  such,  will  be  determined 
from  the  experiments  hereafter  described,  made  by  the  firm  of  Wayss  &  Freytag, 


196 


CONCRETE-STEEL  CONSTRUCTION 


upon  continuous  reinforced  beams.  The  assumptions  on  which  the  success  of  the 
test  rested  were,  that  the  reinforcement  should  conform  to  the  conditions  of  restraint 
and  continuity  and  not  be  arranged  simply  according  to  some  "  System.  "  It 
is  often  forgotten  that  the  elastic  conditions  in  homogeneous  materials  hold 
only  while  strict  propordonality  lasts,  and  that  thus  with  regard  to  the  distribu- 
tion of  stress  at  rupture,  the  same  uncertainty  exists  as  with  reinforced  concrete. 

Reinforced  concrete  beams  and  slabs  can  be  computed  by  the  formulas  for 
continuous  beams  of  constant  section  with  as  much  reason  as  continuous  structural 


Ends  freely  supported. 


Ends  restrained. 

Fig.  i8o. — Maximum  moment  lines  for  continuous  beams  of  three  spans. 


beams  are  designed  on  the  assumption  of  a  constant  moment  of  inertia,  when  the 
area  changes;  that  is,  when  the  maximum  moment  is  employed. 

Computations  will  actually  be  too  favorable,  if,  for  sake  of  simplicity,  the  slabs 
over  the  ribs  and  the  ribs  themselves  over  the  intermediate  columns,  are  considered 
as  freely  supported  (although  really  continuous  members),  and  the  restraint  of 
the  beams  at  the  columns  is  ignored  However,  the  saving  which  exact  computa- 
tions would  make,  could  be  only  insignificant.    The  restraint  of  the  end  panels 


THEORIES  OF  MOMENTS 


197 


of  slabs  at  the  outside  walls  can  only  in  rare  cases  be  secured  through  structural 
measures,  and  is  at  best  very  uncertain,  even  though  the  same  method  is  maintained 
at  the  ends  and  some  reinforcement  bent  upward  in  the  vicinity  of  the  supports. 


i 

i 

— 1  I 

I 

i 

1 

1 

Fig.  i8i. 


The  restraint  afforded  at  the  ends  of  T-beams  which  rest  in  walls  is  even  less. 
If  the  beams  are  supported  by  wall  columns,  a  certain  amount  of  restraining  action 
is  produced,  which  may  be  included  in  computations  under  certain  circumstances. 
The  wall  columns  should  always  be  built  to  resist  bending. 


The  girders  shown  in  Fig.  i8i  support  a  reinforced  concrete  floor  and  are  con- 
tinous  beams  of  two  spans  freely  supported  at  the  ends.  Those  of  Fig.  182  are 
to  be  designed  as  continuous  beams  of  three  spans  with  partial  restraint  of  the 


198 


CONCRETE-STEEL  CONSTRUCTION 


ends  by  the  wall  columns  which  are  built  in  the  outside  walls,  and  support  the 
ends  of  the  beams. 

The  computations  for  a  uniformly  distributed  load  in  the  simplest  form,  from 
the  ordinates  of  the  maximum  moment  line  for  continuous  beams  is  given  by 
Winkler,  in  Vol.  I  of  "  Vortrage  iiber  Briickenbau.  "  His  tables  are  reproduced 
in  the  appendix,  and  are  of  considerable  value  in  connection  with  the  computa- 
tion of  continuous  beams.  The  maximum  moment  line  for  three  spans  is  given 
in  Fig.  1 80.  Of  considerable  value,  also,  are  the  interpolation  tables  for  the 
ready  determination  of  the  influence  lines  for  moments  and  shears  in  continuous 
beams,  prepared  by  Gustav  Griot,  Zurich.  A  further  discussion  of  the  computa- 
tion of  such  moments  will  not  be  given,  since  the  various  methods  are  to  be 
found  in  text-books  of  the  statics  of  building  construction. 

For  many  reasons,  the  "  remnant  "  stresses  in  reinforced  concrete  construc- 
tion are  of  considreable  importance.  It  so  happens  that  the  concrete,  especially 
on  the  tension  side,  undergoes,  by  the  first  loading,  a  certain  permanent  defor- 
mation in  addition  to  the  elastic  one,  which  causes  certain  permanent  stresses 
in  both  materials,  because  of  the  connection  between  the  steel  and  the  concrete. 
These  permanent  stresses  can  but  slightly  influence  the  ultimate  load  through 
repetition,  since  with  repeated  loading,  the  concrete  correspondingly  alters  its 
coefficient  of  elasticity,  which  corresponds  rather  to  the  final  deformation  than 
the  one  produced  by  the  first  application  of  load.  Thus,  when  it  is  stated  that 
reinforced  concrete  beams  which  have  been  subjected  to  bending  have  a  residual 
compression  in  the  lower  concrete,  due  to  the  reaction  of  the  steel  against  the 
permanent  stretch  produced  in  the  concrete  by  the  first  loading,  it  must  also  be 
considered  that  with  repetition  of  load  this  compression  in  the  concrete  invaria- 
ably  must  first  be  overcome  before  tensile  stresses  appear  therein;  that,  further, 
the  concrete  is  not  so  readily  extensible  after  the  first  time;  and  that  the  tensile 
stresses  quickly  increase,  so  that  the  final  value  is  again  approximately  equal  to 
the  first  one.  Aside  from  this  fact,  the  residual  concrete  stresses  would  have 
some  importance  for  the  designer  if  they  exerted  an  influence  upon  the  ultimate 
stresses  in  the  loaded  condition  of  a  beam,  since  he  would  then  have  to 
employ  in  his  designs  a  very  much  lower  resistance  to  load  (Stage  11^  with 
cracked  concrete),  so  as  to  provide  a  proper  factor  of  safety.  However,  these 
"  remnant  "  concrete  strains  and  stresses  have  little  influence  upon  this  condition. 

The  careful  determination  of  the  actual  stresses  in  the  steel  and  concrete 
under  permissible  loads  shows  them  to  correspond  with  those  now  customary  in 
steel  structural  work,  but  this  is  no  reason  why  they  should  be  advocated  for 
reinforced  concrete.  In  an  earlier  chapter  it  was  shown  in  several  cases  that  many 
of  the  methods  adopted  by  custom  in  reinforced  concrete  design,  give  wholly 
erroneous  results;  also  that  regular  steel  construction  was  often  superior  when  one 
could  not  employ  "safe  stresses."  (In  this  connection  may  be  cited,  for  instance, 
the  experiments  of  Schiile  of  Zurich  on  I-beams,  in  which  the  top  flange  failed  by 
lateral  cracking.)  Only  because  designers  of  reinforced  concrete  have  kept 
in  sight  from  the  outset  the  safety  of  their  structures,  keeping  well  within  the 
usual  stresses  under  permissible  loads  wherever  possible,  has  reinforced  concrete 
reached  its  present  development. 

The  shrinkage  of  concrete  produces  secondary  stresses  in  reinforced  concrete 


1 


Fig.  1S3. — Continuous  Test  Beam  I  with  haunches.    Breaking  load  34.4  t.  (37. S  tons). 


'  t  i  f  }    I  I,   *  .1    t   s  ~T~~^ 


Llll 

1 

Fig.  184— Continuous  Test  Beam  II  without  haunches  but  with  spirals  and  shear  rods.    Breaking  load  31.9  t.  (35.1  tons.) 


% 

-A 

i   

 "  —  ■  S.SS 

Fig.  185.    Continuous  Test  Beam  III,  without  haunches  but  with  spirals;  reinforcement  same  as  Beam  II  except  for  shear  rods  and  the  substitution  of  a  20  mm.  (}  in.  approx.)  rod  for  one  14  mm.  (}  in.  appros.)  in  diameter. 

Breaking  load  25.4  t.  (28.0  tons). 

To  face  page  100 


CONTINUOUS  T-BEAM  EXPERIMENTS 


199 


structures.  The  amount  of  this  shrinkage  bears  a  definite  relation  to  the  pro- 
portions of  the  mixture.  In  long  structures  allowance  must  be  made  for 
expansion  and  contraction  from  temperature  change,  by  means  of  expansion 
joints  of  suitable  size  (20  to  40  mm.)  (J  to  in.  approx.).  In  high  buildings, 
the  joints  can  be  carried  either  through  the  centers  of  beams  and  columns  or 
through  the  panels.  The  latter  arrangement  leads  to  a  cantilever  construction 
of  the  floor  and  beams.  The  expansion  joint  can  be  constructed  without  any 
open  space,  since  it  tends  only  to  open  because  of  the  shrinkage  of  the  concrete. 
Buildings  provided  with  such  joints  actually  remain  free  from  cracks,  while 
otherwise  the  danger  of  cracks  occurring  at  undesirable  points  always  exists. 


EXPERIMENTS  WITH  CONTINUOUS  T=BEAMS 

Since  experiments  with  properly  constructed  continuous  T-beams  were 
unknown,  and  because  of  the  importance  of  the  subject,  the  firm  of  Wayss  and 
Freytag  carried  out  tests  of  three  beams  of  T-section  in  accordance  with  plans 
prepared  by  the  author.  The  specimens  are  illustrated  in  Figs.  183-185.  They 
consisted  of  continuous  beams  of  two  spans  each  5.9  m.  (19.4  ft.)  wide,  the  ribs 
of  the  beams  being  14  cm.  (5.5  in.)  broad  and  25  cm.  (9.8  in.)  high,  above  which 
was  a  slab  i  m.  (39.4  in.)  wide  and  10  cm.  (3.9  in.)  thick.  For  sake  of  stability, 
lateral  ribs  were  built  over  the  supports.  The  arrangement  of  the  reinforce- 
ment was  made  in  accordance  with  the  moment  line  for  uniformly  distributed 
loading  over  two  openings.  The  specimens  were  five  weeks  old  at  the  time 
of  the  test. 

Beam  I  showed  the  usual  arrangement  of  continuous  reinforced  concrete 
beams,  with  haunches  at  the  center  support.  Since  the  dead  load  amounted 
to  325  kg  m  (218  lbs/ft)  a  five  load  of  equal  amount  would  produce  a 
total  of  650  kg/m  (436  lbs.  per  running  ft.)  under  which  the  reaction  at 
the  left  support  would  be  (2  =  650X6.15X1  =  1500  kg.  (3300  lbs.),  so  that  tq 
=  3.6  kg/cm^  (51  lbs/in^).  Since  the  bent  reinforcement  was  ample  at  that 
point  to  resist  the  diagonal  tension,  the  adhesion  could  be  computed  according 

to  the  formula  ti  =    ^     and  this  gave  "1  =  2.85  kg/cm^  (41  lbs/in-),  so  that  a 

2Z  U 

large  factor  of  safety  was  secured  at  the  ends. 

At  0.4/  the  bending  moment  under  the  same  load  would  be 

i\/=o.o7X65oX6. 152x100  =  172090  cm.-kg.  (153385  in.-lbs.) 

and  consequently, 

(7e  =  iooo  kg/cm^  (14223  lbs/in2),    and    (T6  =  i7-4  kg/cm^  (247  lbs/in^). 
Over  the  center  pier  the  computed  stresses  were 

(Te  =  990  kg/cm2  (14081  lbs/in^),    and    ^7^  =  45. 7  kg/cm^  (649  lbs/ in^). 


200 


CONCRETE-STEEL  CONSTRUCTION 


At  a  total  load  of  6t.  (6.6  tons)  on  both  spans,  the  first  cracks  had  already 
appeared  in  the  region  of  the  greatest  positive  moments,  and  at  i4t.  (15.4  tons) 
they  showed  themselves  at  the  points  of  negative  moment.  With  increase  of  load, 
somewhat  later  slightly  diagonal  cracks  followed  in  the  outer  zones  of  positive 
and  negative  moment.  Between  these  two  regions  at  points  about  jl  from  the 
end,  where  the  moment  was  zero,  a  space  in  the  beam  remained  with  no  cracks 
up  to  the  ultimate  load.  At  those  points,  the  principal  stresses  were  of  shear, 
which  were  carried  by  the  bent  rods,  according  to  the  computations  made  above. 
From  the  distribution  of  cracks,  which  are  shown  for  only  one  span  in  Fig.  183, 
it  is  clear  that  a  theoretically  continuous  reinforced  concrete  beam  which  is 
actually  so  constructed  will  act  as  such. 

Rupture  resulted  at  a  load  of  34. 4t.  (37.8  tons),  from  crushing  of  the  concrete 
on  the  under  side  of  the  haunches.  This  occurred  at  a  point  where  a  stirrup 
had  been  displaced  by  the  ramming  of  the  concrete,  so  that  the  stirrup  spacing 
was  24  cm.  (9.4  in.)  as  that  point,  instead  ot  15  cm.  (5.9  in.).  The  two  round 
rods  of  10  mm.  (fin.)  diameter,  were  buckled  in  consequence,  in  the  manner 
shown  by  the  photograph  in  Fig.  186.    A  compression  reinforcement  thus  shows 


Fig.  186. — Beam  I,  Cracks  and  Rupture  in  the  vicinity  of  the  intermediate  support. 


itself  of  value,  when  it  is  prevented  from  buckling  by  closely  placed  stirrups, 
but  otherwise  it  may  be  an  actual  detriment.  Simultaneously,  with  the  break 
at  the  center  support,  one  also  took  place  at  0.4I,  since  with  the  giving  way  of  the 
haunch  the  positive  moment  was  considerably  increased  so  that  failure  naturally 
resulted  at  that  point. 

For  the  ultimate  load  of  34. 4t.  (37.8  tons),  when  the  dead  weight  is  also  con- 
sidered, the  computations  give 

To  =  i7.3  kg/cm2  (246  lbs/in2), 
Ti  =  i3.7  kg/cm2  (195  lbs/in2). 

At  0.4I,  according  to  Stage  116,  there  results 

(Te=48oo  kg/cm^  (68,273  lbs/in^), 
^7^  =  83. 5  kg/cm2  (1188  lbs/in2). 


CONTINUOUS  T-BEAM  EXPERIMENTS 


201 


When  the  steel  stress  is  computed  at  the  time  of  rupture  with  the  arm  of  the 
couple  at  32  cm.  (12.6  in.),  it  is  found  to  be  4450  kg/cm^  (63,294  lbs/in-).  At 
the  point  where  the  concrete  was  crushed  on  the  haunch,  the  moment  of  the 
continuous  beam  (computed  for  a  constant  section)  equals  7.0  m-t.  (25.3  ft. -tons). 
From  this  there  results 

(7c  =  3700  kg/cm^  (43,626  ll)s/in2), 
(76  =  171  kg/cm^  (2432  lbs/in^), 

so  that  the  destruction  of  the  concrete  at  this  point  is  adequately  explained. 
If  it  is  further  considered  that  the  compression  acted  parallel  to  the  under  side, 
and  that  the  stress  computed  for  a  vertical  section  is  to  be  divided  by  cos^x,  a 
compression  on  the  concrete  of  184  kg/cm^  (2574  lbs/in^)  is  obtained.  The 
shearing  stress  computed  for  the  section  which  failed,  according  to  the  formula 

Q  iM 

bTo  =  - — ---^  tana, 

Z  OS" 

amounts  to  70  =  9.9  kg/cm^  (141  lbs/in^). 

The  horizontal  crack  at  the  point  of  rupture  should  be  noted.  At  28. 6t. 
(31.5  tons)  it  was  already  visible  and  corresponded  to  a  shearing  of  the  concrete. 
At  the  front  end  of  the  haunch,  the  compression  in  the  concrete  acted  horizon- 
tally, and  because  of  the  abrupt  change  of  direction  of  the  lower  edge,  the  concrete 
of  the  haunch  could  only  be  affected  by  horizontal  shearing  stresses  in  a  way  to 
alter  the  [direction  of  the. — Trans.]  compressive  forces,,  Since  the  change  is  abrupt, 
the  principal  stresses  cannot  develop  at  that  point,  and  moreover  the  transfer 
may  be  supposed  to  take  place  through  a  sort  of  toothing,  as  in  the  case  of  plain 
shear  (Fig.  33).  The  danger  of  this  horizontal  shear  is  the  greater,  the  steeper 
is  the  haunch.    Flat  and  especially  rounded  transitions  are  therefore  preferable. 

Beam  II  had  no  haunches,  the  upper  reinforcement  at  the  center  support 
being  increased  over  that  of  Beam  I  so  as  to  give  equal  carrying  power,  and 
the  lower  side  was  strengthened  against  compression  by  a  spiral  of  10  mm. 
(f  in.)  round  steel,  with  a  diameter  of  10  cm.  (3.9  in.)  and  a  pitch  of  3  cm.  (1.2  in). 
Since  the  shearing  stresses  were  greater  because  of  the  absence  of  a  haunch, 
two  extra  shear-rods  were  introduced  (Fig.  184). 

The  distribution  of  cracks  was  just  the  same  as  in  the  foregoing  beam,  only 
the  point  of  zero  moment  occurred  somewhat  nearer  the  center  support  (which 
is  in  accord  with  theory),  because  the  larger  section  near  the  center  pier  of 
Beam  I  would  move  the  point  of  zero  moment  further  into  the  span.  The 
stresses  computed  for  the  section  with  maximum  positive  moment  at  the  load 
of  31.9  t.  (35.1  tons)  which  produced  rupture  of  the  concrete  on  the  lower  side 
near  the  center  support,  were 

(je=425o  kg/cm2  (60,449  lbs/in2). 


(7^  =  76.6  kg/cm2  (1089  lbs/in2). 


202 


CONCRETE-STEEL  CONSTRUCTION 


Since  failure  occurred  in  this  section  at  the  same  time  with  that  of  the  concrete 
at  the  center  support,  Oe  was  relatively  the  same  at  the  two  points,  just  as  in 
Beam  I.  When  the  bending  moment  and  corresponding  stresses  are  computed 
for  the  point  of  failure  located  about  0.3  m.  (0.98  ft.)  from  the  axis  of  the  center 
support,  there  result  at  the  instant  of  failure  ^7^,= 33 15  kg/cm^  (47,150  lbs/in^), 
and  (7^  =  311  kg/ cm^  (4423  lbs/ in^).  Because  of  the  50  cm.  (19.7  in.)  breadth 
of  the  supports,  a  certain  added  security  was  attained  at  those  points,  so  that 
these  figures  are  rather  too  small  than  too  large. 

The  diagonal  cracks  cutting  the  shear  rods,  were  apparent  at  a  load  of  16  t. 
(17.6  tons),  corresponding  to  a  shearing  stress  of  to  =  i5.5  kg/cm^  (220  lbs/in^). 
There,  the  vertical  pressures  between  the  concrete  layers  produced  by  the  reaction 
of  the  support,  acted  to  delay  the  appearance  of  the  first  shear  crack,  as  did 


Fig.  187. — Beam  II,  Cracks  and  Rupture  in  the  vicinity  of  the  intermediate  support, 

also  the  shear  rods  and  the  stirrups.  The  adhesion  must  be  computed  at  those 
points  for  the  circumference  of  the  upper  reinforcement,  and  was  6.1  kg/cm^  (87 

lbs/in2)      failure,  according  to  the  formula  '^^^'^Jf-  Since  in  continuous  beams, 

an  abundance  of  steel  is  to  be  found  over  the  intermediate  supports,  the  adhesion 
there  will  always  be  found  low  in  computed  value. 

Beam  III  was  constructed  exactly  like  Beam  II,  except  that  the  two  14  mm. 
in.)  shear  rods  were  omitted  and  the  upper  14  mm.  reinforcing  rod  was 
increased  to  20  mm.  (|  in.).  The  distribution  of  cracks  as  shown  in  Fig.  185 
was  hke  that  of  Beam  II.  Failure  again  occurred  through  rupture  of  the  lower 
concrete  near  the  center  support,  while  failure  also  took  place  at  the  point  of 
maximum  positive  moment  at  the  same  time  that  the  ultimate  carrying  power 
was  exceeded  at  the  other  point.  With  the  failure  of  the  concrete  on  the  under 
side,  indications  also  existed  of  a  shearing  of  the  concrete  in  a  horizontal  direc- 
tion just  above  the  spiral  (Fig.  188).  Since  the  breaking  load  reached  only  25.4  t. 
(27.9  tons),  when  compared  with  Beam  II  the  utiHty  of  special  shear  rods  is 
disclosed.  These  may  be  unnecessary  when  the  main  reinforcement  can  be 
bent  at  the  desired  points  without  reducing  its  area  below  that  clearly  necessary 
to  resist  the  negative  moments. 


CONTINUOUS  T-BEAM  EXPERIMENTS 


203 


Experiments  with  continuous  beams  are  more  difficult  to  execute  than  those 
on  simple  beams,  since  an  inequality  in  the  loading  affects  the  results.  The 
firm  of  Wayss  &  Freytag  will  repeat  these  experiments  with  somewhat  heavier 
reinforcement  near  the  ends,  and  also  make  two  other  beams  without  spirals 
or  haunches. 

From  the  foregoing  results,  the  effect  of  lack  of  haunches  in  continuous  rein- 
forced concrete  beams  can  be  seen.    Their  economic  value  is  shown  in  the  follow- 


FiG.  i88. — Beam  III,  Cracks  and  Rupture  in  the  vicinity  of  the  intermediate  supports. 

ing  table,  which  gives  the  amount  of  the  main  reinforcement  including  the 
spirals  but  omitting  the  stirrups: 


Beam 
I 
II 
III 


Ultimate  Load 
34-4  t. 
31-9 
25-4 


Reinforcement 

75-3  kg. 

90.0 

91.0 


From  these  experiments,  especially  from  Beam  I,  it  may  be  concluded  that 
reinforced  concrete  beams  can  be  constructed  as  continuous  members.  It  is 
believed  that  they  should  almost  always  be  so  designed  when  it  is  considered 
that  all  parts  of  a  building  of  this  type  are  monolithic. 

This  also  necessarily  applies  to  floor  slabs  carried  by  reinforced  concrete 
beams,  which  must  withstand  the  negative  moments  over  the  latter,  for  since 
they  act  as  the  compression  chords  of  the  latter,  they  should  not  be  allowed  to 
become  cracked  on  the  upper  side  next  the  beams. 


PART  II 


CHAPTER  XII 
APPLICATIONS  OF  REINFORCED  CONCRETE 
HISTORICAL 

Joseph  Monier,  who  made  the  first  practical  use  of  it  in  1868,  is  usually 
credited  with  being  the  inventor  of  reinforced  concrete.  However,  traces  of 
this  method  of  construction  are  found  at  an  earlier  period.  For  instance,  at  the 
Paris  Exposition  of  1855,  Lambot  exhibited  a  boat  made  of  reinforced  concrete. 
At  the  International  Exposition  of  1867,  besides  the  better  known  Monier, 
Francois  Coignet  was  represented.  Since  i860  he  had  designed  floors,  arches 
and  pipes,  in  the  construction  of  which  the  fundamental  principles  of  reinforced 
concrete  construction  are  recognizable. 

To  Monier,  however,  belongs  the  credit  of  having  devoted  himself  to  the 
new  method  of  construction  with  perseverance  and  success. 

Originally,  he  was  the  owner  of  an  important  nursery  in  Paris.  His  first 
attempts  were  to  make  large  plant  tubs  which  would  be  more  durable  than  those 
of  wood,  and  more  readily  transportable  than  those  of  cement.  He  sought  to 
attain  his  object  by  introducing  iron  rods  of  small  diameter  into  the  cement 
sides  of  the  tubs,  and  extended  this  method  of  construction  to  the  production  of 
large  water  tanks.  In  1867  he  took  out  his  first  French  patent,  which  he  soon 
followed  with  a  large  number  of  others  on  reservoirs,  floors,  straight  and  arched 
beams  in  combination  with  floors,  etc.  In  his  patent  drawings  are  already  dis- 
closed all  the  elements  which  are  to-day  employed  in  the  various  construction 
details  of  the  vairous  systems. 

It  is  easy  to  understand  how  Monier's  invention,  but  little  understood  and 
based  as  it  was  on  an  entirely  empirical  foundation,  was  destined  to  develop  along 
entirely  different  lines  in  the  hands  of  engineers. 

In  1884  the  so-called  Monier  patents  were  purchased  by  the  firms  of  Freitag 
and  Heidschuch  in  Neustadt-on-the-Haardt,  and  of  Martenstein  and  Josseaux  in 
Offenbach-on-the-Main.  The  first  mentioned  firm  acquired  the  rights  for  all 
of  South  Germany,  with  the  exception  of  Frankfort-on-the-Main,  and  vicinity, 
which  territory  was  reserved  to  themselves  by  the  last  mentioned  firm.  At  the 
same  time  both  parties  acquired  from  Monier  options  for  the  whole  of  Germany, 
which,  however,  they  rehnquished  a  year  later  to  engineer  Wayss.  The  latter, 
in  connection  with  the  above  mentioned  firms,  conducted  load  tests  in  Berlin, 
the  results  of  which  were  made  pubHc  in  1887,  in  the  brochure  "  Das  System 
Monier,  Eisengerippe  mit  Zementumhullung,"  on  the  basis  of  which  Wayss 
succeeded  in  introducing  the  Monier  system  into  public  and  private  edifices. 

204 


HISTORY 


205 


In  that  pamphlet,  Wayss  first  expressed  the  decided  opinion  that  the  steel  in 
reinforced  concrete  construction  must  be  placed  where  the  tensile  stresses  occur. 
He  perceived  that,  owing  to  the  extraordinary  adhesion  of  cement  concrete  to 
iron,  both  elements  must  act  together  statically,  and  found  his  theory  confirmed 
by  his  numerous  tests.  The  Wayss  experiments  included  not  only  strength 
tests  of  all  kinds,  but  were  also  extended  to  include  tests  of  protection  against 
fire,  and  protection  of  the  embedded  steel  against  rust,  as  well  as  of  the  adhesion 
of  iron  to  concrete. 

Examples  were  also  given  in  the  above  mentioned  pamphlet  explaining 
the  commercial  praticability  of  the  new  method  of  construction  compared  with 
the  older  systems.  The  great  capacity  of  the  Monier  slabs  to  resist  shock  had 
at  that  time  already  been  demonstrated.  The  tests  were  witnessed  by  official 
representatives  and  private  engineers  and  architects.  Government  Architect 
Koenen,  now  Director  of  the  Actiengesellshaft  fur  Beton-  und  Monierbauten, 
in  Berlin,  was  commissioned  by  Wayss  to  deduce  methods  of  computation  from 
these  tests,  which  latter  were  published  in  the  same  pamphlet  and  also  in  the 
volume  of  the  "  Zentralblatter  der  Bauverwaltung  "  for  1886. 

Commencing  at  that  time,  a  theoretical  foundation  was  evolved,  according 
to  which  the  design  of  reinforced  concrete  work  could  be  effected,  and  through 
these  preliminary  labors,  this  method  of  construction  was  extensively  adopted 
in  Germany  and  Austria.  A  turning  point  in  its  development  was  the  Interna- 
tional Exposition  in  Paris,  in  1900,  and  the  report  by  Emperger,  published  at 
that  time  in  regard  to  the  position  which  the  subject  occupied. 

Because  of  the  scientific  investigation  of  reinforced  concrete  during  the 
past  few  years,  it  has  made  rapid  progress  in  Germany.  It  was  specially  pro- 
moted by  the  publication,  in  1904,  through  the  cooperation  of  experts  and  prac- 
tical men  of  the  "Leitsatze"  of  the  Verbands  Deutscher  Architekten-  und 
Ingenieurvereine  and  the  Deutschen  Betonvereins,  as  well  as  by  the  "  Regulations" 
of  the  Prussian  Government,  which  abolished  many  restrictive  rules,  cleared 
the  way,  and  inspired  in  the  widest  circles  confidence  in  the  new  method  of 
building. 

At  the  present  time,  in  many  countries,  commissions  are  investigating  the 
subject  of  reinforced  concrete.  The  French  Commission  has  already  completed 
its  labors  and  pubhshed  its  findings  in  a  special  report.  The  German  Commit- 
tee on  reinforced  concrete  has  arranged  an  extensive  programme,  which  is  being 
executed  in  numerous  testing  laboratories.  Besides  this  committee,  there  is 
the  Reinforced  Concrete  Commission  of  the  Jubilee  of  the  Foundation  of  German 
Industries,  which  continues  in  existence,  and  some  of  the  important  results  of  the 
activities  of  which  have  been  given  in  the  foregoing  matter.  The  Swiss  Com- 
mission will  complete  its  work  during  the  next  year,  and  make  suggestions  for 
the  construction  and  design  of  reinforced  concrete  structures. 

In  the  United  States  reinforced  concrete  has  been  employed  for  a  considerable 
time,  but  the  wide  difference  among  the  systems  employed,  and  the  lack  of 
method  in  their  preparation,  have  prevented  rational  development.  The  Ran- 
som, Wilson,  expanded  metal,  etc.,  systems,  have  found  wide  adoption  in  build- 
ings in  that  country.  The  Melan  system  was  introduced  into  America  by  F. 
von  Emperger,  and  came  into  extensive  use  in  bridge  building. 


206 


CONCRETE-STEEL  CONSTRUCTION 


In  addition  to  the  Monier  system,  which  was  widely  developed  in  France 
(the  land  of  its  origin),  a  large  number  of  other  systems  arose  there,  of  which 
only  the  names — Cottancin,  Bordenave,  Coignet,  Bonna,  Latrai,  etc. — can  be 
mentioned.  The  best  known  system  is  that  of  Hennebique,  which,  at  the  start, 
like  those  of  his  English  and  American  co-laborers,  aimed  chiefly  at  security 
against  fire.  It  is  extensively  employed  in  France,  Belgium,  and  Italy.  Henne- 
bique's  ideas  of  construction  were  not  altogether  new,  perhaps,  being  contained 
in  part  in  the  Monier  patent  specifications.    Thus,  are  found,  beams  reinforced 


Fig.  189. — Monier's  Reinforced  Concrete  Patent  Drawings  of  1878. 


with  similar  heavy  round  rods  and  wide  stirrups,  bent  rods  used  in  floors,  beams,, 
etc.  The  first  reinforced  concrete  beams  in  combination  with  slabs  occurred 
as  early  as  1886,  in  connection  with  the  erection  of  the  library  in  Amsterdam. 
Since  1892,  in  chronological  order,  there  followed  Coignet,  Sanders,  Ransome, 
and  then  Hennebique. 

Of  the  better  known  systems  for  floor  slabs  and  beams,  the  following  may  be 
enumerated:  The  Monier  system  (Fig.  190)  employs  numerous  distributing 
rods  at  right  angles  to  the  supporting  rods,  the  two  being  wired  together  at  the 
points  of  intersection.    Formerly,  the  distributing  rods  crossing  the  supporting 


Fig.  190. — Monier  System.  FiG.  191. — Hyatt  System. 


ones  were  regarded  as  a  means  of  preventing  the  concrete  from  slipping  length- 
wise of  the  supporting  rods.  When  it  was  realized,  however,  that  the  adhesion 
was  sufficient  for  this  purpose,  the  distributing  rods  were  spaced  further  apart. 
The  Monier  system,  or  the  ordinary  grid  of  round  rods  spaced  6  to  10  cm.  (2.4 
to  4  in.)  apart,  found  extensive  employment  in  the  construction  of  reservoirs  of 
every  description. 

Hyatt  (Fig.  191)  made  the  supporting  rods  of  flat  iron  laid  on  edge,  in  which 
holes  v/ere  punched,  through  which  were  passed  distrubuting  rods,  made  of 
small  round  iron. 

The  Ransome  system.  Fig.  192,  which  attained  considerable  importance  in 
America,  suppresses  the  distributing  rods  entirely,  and  uses  for  the  supporting 


SYSTEMS 


207 


rods,  spirally-twisted  square  iron,  to  prevent  any  slipping  in  the  concrete.  Other 
inventors,  like  Cottacin  (Fig.  193)  have  woven  supporting  and  distributing  rods 
together,  to  form  a  rectangular  network. 

For  continuous  floors  with  steel  beams,  bent  rods  in  the  form  of  flat  iron, 
to  which  are  riveted  angle  iron  clips,  are  used  in  the  Klett  system  (Fig.  195) 


Fig.  192, — Ransome  System. 


Fig.  193. — Cottacin  System. 


and  the  Wilson  and  Koenen  systems  (Fig.  194).  In  the  two  last-named  systems, 
the  slab  is  made  heavier  at  the  supports,  as  required  by  the  greater  moment  of 
a  continuous  slab  at  those  points.  For  varying  loads,  however,  a  single  reinforce- 
ment must  be  regarded  as  inadequate. 


Fig.  194. 
Koenen  System. 


Fig.  195. 
Klett  System. 


The  Matrai  system  incloses  in  the  concrete  a  network  of  wires  suspended, 
chain-like,  between  iron  supports,  crossing  one  another  so  as  to  form  squares. 

In  the  Hennebique  system  (Figs.  196,  197),  the '  reinforcement  of  slabs  and 
beams  consists  of  two  series  of  rods.  One  series  is  straight  and  lie  in  the  lower 
part  of  the  concrete.  Over  the  supports  are  top  rods  which  are  bent  down  near 
the  center  of  the  span  and  finally  He  close  to  the  straight  ones. 


Fig.  196. 
Slab  of  the  Hennebique  System. 


Fig.  197. 

Beam  of  the  Hennet^ique  System. 


Fig.  198. 
Flat-iron 
Stirrup. 


Hennebique  rightly  recognizes  in  the  bent  rods  a  preventive  of  shearing 
strains  and  uses  them  also  with  slabs  and  beams  which  are  not  restrained.  He 
also  uses  flat  iron  stirrups  in  slabs  and  beams,  of  the  form  shown  in  Fig.  198. 

Hennebique  is  entitled  to  the  credit  of  having  introduced  into  construction 
work,  on  a  large  scale,  reinforced  concrete  beams  and  columns,  and  of  having 
developed  new  fields  for  reinforced  concrete  work. 

The  systems  of  Klett  and  Wilson,  used  for  floor  slabs,  in  which  the  reinforce- 
ment consists  exclusively  of  suspended,  bent,  flat  iron,  have  their  counterpart 


208  CONCRETE-STEEL  CONSTRUCTION 


in  the  beam  construction  of  the  Moller  system  (Fig.  199)  used  for  the  construc- 
tion of  bridges  of  spans  up  to  20  m.  (66  ft.).  In  this  case,  the  ribs  are  reinforced 
on  the  lower  side  by  suspended,  bent,  flat  irons  that  are  anchored  over  the  points 
of  support,  and  on  which  angle  iron  clips  are  riveted.  The  ribs  have  the  fish 
belly  shape,  and  follow  exactly  the  line  of  the  suspended  rods;  their  depth 
decreases  as  they  approach  the  points  of  support,  while  the  thickness  of  the  floor 
slab  is  increased  at  those  points.  The  floor  must  take  care  of  the  longitudinal 
pull  of  the  hanging  flat  irons,  and  is  reinforced  at  right  angles  to  their  length 
by  small  I-beams  or  angle  irons. 


Fig.  199. — T-beam  Construction  of  the  Moller  System. 


The  efforts  made  to  manufaacture  the  structural  parts  of  a  reinforced  con- 
crete structure  at  a  special  plant,  and  then  transport  them  in  finished  state  to  the 
building,  have  produced  a  number  of  floor  systems.  The  most  successful  in  this 
respect  are  the  hollow  beams  of  the  Siegwart  system,  which  form  a  continuous 
floor  when  laid  close  together,  and  the  beams  of  the  Visintini  system*  which  are 
used  in  the  same  manner,  and  consist  of  upper  and  lower  transverse  members 
with  connecting  webs. 

The  theory  of  reinforced  concrete  construction  has  undergone  many  changes. 
The  first  suggestions  of  the  prevailing  methods  of  computation  as  given  in  the 
''Leitsatze"  are  to  be  found  in  a  communication  from  Coignet  and  de  Tedesco 
in  1894,  where,  for  instance,  the  quadratic  equation  is  given  as  the  means  of 
determining  the  distance  of  the  neutral  layer  from  the  upper  surface  of  the  slab. 
This  publication  is  little  known  and  consequently  these  formulas  were  independ- 
ently discovered  by  various  authors,  by  Ritter  of  Zurich,  for  instance,  in  1899, 
and  Emperger.  The  centroid  of  compression  was  also  first  located  in  the  center 
of  the  pressure  zone,  in  place  of  at  two-thirds  of  its  height. 

In  the  first  edition  of  the  work  of  Christophe,  Annale  des  Travaux  publics 
de  Belgique,"  1899,  the  theory  to-day  contained  in  the  "Leitsatze"  is  completely 
discussed,  including  the  provision  that  the  concrete  can  withstand  no  tension. 

Autenrieth  of  Stuttgart  contributed  to  Vol.  XXXI,  1887,  of  the  Zeitschrift 
des  Vereins  Deutscher  Ingenieure,^  ''a  graphical  method  of  calculating  the 
anchors  which  fasten  floors  to  plane  surfaces."  The  methods  there  set  forth  are 
identical  with  those  herein  employed  for  the  calculation  of  reinforced  concrete 


*  Beton  und  Eisen,  No.  Ill,  1903. 

t  "Berechnung  der  Anker,  welche  zur  Befestigung  von  Flatten  an  ebenen  Flachen  dienen.*' 


BUILDINGS 


209 


work,  so  that  the  processes  indicated  for  simple  bending  and  for  flexure  with  axial 
pressure  have  been  carried  over,  without  change,  to  reinforced  concrete  work. 
This  graphical  method  has  been  reproduced  on  page  130  et  seep,  and  it  is 
recommended  for  use  in  connection  with  complicated  forms  of  cross-section. 

BUILDINGS 

In  buildings,  reinforced  concrete  may  be  used  only  for  floors  between  steel 
beams,  or  in  monolithic  construction  for  beams  and  columns  as  well. 

Reinforced  concrete  floors  between  I-beams  were  fomerly  constructed  as 
Monier  slabs,  resting  freely  on  the  lower  flanges  of  the  beams,  or  in  the  form  of 
flat  plates  carried  continuously  over  the  upper  flanges.  The  present  customary 
method  of  constructing  continuous  concrete  floors  between  I-beams  is  shown 


u 


Fig.  200. — Holzer  System. 


in  Fig.  3.  The  lower  flanges  of  the  beams  are  wrapped  with  woven  wire,  so  that 
they  wiU  take  the  ceihng  finish  which  also  protects  the  metal  to  some  extent 
from  the  direct  effects  of  fire  in  case  of  a  conflagration.  Among  the  systems 
using  reinforced  concrete  between  I-beams,  the  foUowing  may  be  enumerated: 
Floors  according  to  the  Holzer  system  (Fig.  200).  This  belongs  to  the  class 
of  simple  reinforced  concrete  slabs  with  free  ends,  and  consists  of  a  reinforce- 
ment of  small  beams  22  mm.  (|  in.)  deep.     The  only  value  of  this  special  form 


Fig.  201. — Zollner  Cellular  Floor. 


of  section  inheres  in  the  possibility  of  erecting  the  floors  without  wooden  forms, 
the  small  beams  not  possessing  any  increased  carrying  capacity  in  comparison 
with  round  rods  of  equal  size.  The  support  for  the  concrete  is  a  cane  mat, 
stiffened  by  round  rods  and  suspended  from  the  I-beams  by  wires.  The  mats 
are  thus  made  to  support  the  load  of  the  floor  during  construction.  The  usual 
spanss  are  from  i.om.  to  a  maximum  of  2.5  m.  (3  to  8  ft.  approximately). 

The  Zollner  ceUular  floor  system  (Fig.  201)  is  suited  to  wider  spans,  about 
4.5  to  7  m.  (15  to  23  ft.  approx.).    With  comparatively  small  dead  weight,  it 


210 


CONCRETE-STEEL  CONSTRUCTION 


possesses  considerable  structural  depth,  and  when  set  between  the  lower  flanges 
of  I-beams,  requires  but  little  filling.  The  great  depth  and  low  weight  are 
obtained  by  the  use  of  a  series  of  hollow  blocks,  made  from  burned  clay.  The 
actual  supporting  structure  of  the  floor  consists  of  T-beams  of  concrete,  arranged 
side  by  side,  the  stems  of  which  carry  the  reinforcement.  Instead  of  hollow 
blocks,  a  light  cinder  concrete  may  be  used.  It  is  also  possible  to  construct  a 
continuous  cellular  floor  between  I-beams  or  reinforced  concrete  ribs.* 

For  sake  of  completeness  there  may  also  be  mentioned  the  Monier  arch  exten- 
sively employed  during  the  early  periods  of  reinforced  concrete  work,  but  now 
replaced  by  more  practical  systems.  The  floor  systems  designed  to  be  used  with 
reinforced  concrete  members  already  in  place,  are  very  numerous  and  their 
enumeration  would  be  too  lenghty. 

Monolithic  System  of  Reinforced  Concrete.  Of  far  greater  importance 
than  the  various  methods  of  floor-slab  construction  between  steel  beams,  are  the 
complete  reinforced  concrete  buildings,  in  which  all  the  load  sustaining  parts 


Fig.  202. — Warehouse  for  the  Government  Railway  at  Elberfeld  in  Opliden. 


(floors,  beams  and  columns)  are  executed  in  reinforced  concrete.  This  mate- 
rial is  best  adapted  for  long  span,  heavily  loaded  floors,  and  consequently,  advanta- 
geously replaces  the  usual  building  materials  for  all  factories,  warehouses,  etc. 

All  parts  are  made  in  situ,  so  that  the  entire  frame  is  of  a  perfectly  rigid,  mono- 
lithic character.  The  columns  are  rigidly  joined  with  the  beams,  and  the  joint 
is  given  additional  strength  by  the  haunches  under  the  beams.  By  this  means, 
even  with  thin  enclosing  wafls,  the  stabiHty  of  the  entire  structure  against  lateral 
forces  is  considerably  increased,  as  compared  with  those  employing  steel  beams 
and  columns.  In  the  latter  there  is  always  a  certain  amount  of  joint-like  mobility 
in  the  connections.  Reinforced  concrete  beams  may  be  supported  directly  on 
the  outside  walls,  when  of  ordinary  masonry,  but  in  order  to  prevent  settlement 
the  walls  must  have  a  very  solid  foundation  and  should  be  laid  in  cement  mortar. 
Or,  wall  columns  may  be  carried  up  to  receive  the  floor  loads  transmitted  by 


*  Beton  und  Eisen,  No.  Ill,  1903. 


BUILDINGS 


211 


Fig.  204. — Reinforced  Concrete  Construction  of  Floors  and  Columns  in  the  Speyer  Cotton 

Mill  on  the  Rhine. 


212 


CONCRETE-STEEL  CONSTRUCTION 


.  the  main  beams.  If  the  wall  columns  are  then  connected  with  special  wall 
beams,  which  are  made  to  support  the  floors  at  the  outside  walls,  the  load  carry- 
ing reinforced  concrete  frame  can  be  reared  by  itself,  and  completed  independently 
of  all  w^all  work.  When  relieved  alike  of  the  structural  weight  of  floors  and 
their  live  loads  the  masonry  of  the  enclosing  walls  becomes  simply  a  covering 
that  serves  merely  to  impart  to  the  building  the  customary  appearance.  (Fig. 
202).  But  when  an  outside  wall  is  erected  of  such  a  variety,  revealing  nothing  of 
the  particular  interior  construction,  it  can  be  built  much  lighter  than  could 
otherwise  be  allowed  or  even  than  the  building  laws  prescribe.  It  is  then  possible, 
and  this  is  of  the  greatest  importance  in  factory  construction,  to  leave  very 
large  opeings  for  the  admission  of  light.  For  this  purpose,  the  whole  height 
to  the  lower  edge  of  the  floor  slab  may  be  employed,  since  the  ribs  of  the  wall 
beams  and  window  lintels  can  be  carried  above  the  floors  just  as  well  as  below 
them. 

The  wall  columns  are  subjected  to  certain  bending  stresses  from  the  load- 
ing oE  the  girders  and  because  of  their  rigid  connection  with  them,  and  in  conse- 
quence are  usually  carried  up  of  rectangular  section, 
with  the  long  side  parallel  to  the  length  of  the  girder. 

For  some  time,  steel  columns  and  beams  have  not 
been  considered  fire-proof  building  material.  Columns 
bend  at  600  to  800°  C.  (iioo  to  1400°  F.).  while 
beams  break  or  push  apart  the  outside  walls,  by  their 
expansion.  An  instructive  picture  of  the  destruction 
to  which  a  steel  structure  is  exposed  in  case  of  a 
conflagration,  is  the  view  shown  in  Fig.  203,  of  the 
upper  story  of  the  Speyer  woolen  factory  after  a 
fire. 

The  interior  of  the  same  story,  rebuilt  in  rein- 
forced concrete,  is  also  shown  in  Fig.  204.    The  concrete 
columns  stand  on  the  cast  iron  ones  of  the  lower 
story,  which  were  spared  by  the  fire  (Fig.  205).  To 
insure   better   insulation  and  reduce  the  weight  on 
the  old  columns  as  much  as  possible,  the  floor  slabs 
were  made  of  pumice-stone  concrete,  while  Rhine  gravel 
was  used  for  the  concrete  in  the  beams  and  columns. 
In  distinction  from  steel  construction,  buildings  of  reinforced  concrete  are 
fire-proof,  for  in  them  the  metal  does  not  play  as  important  a  part,  and  moreover 
it  is  effectively  protected  from  the  effects  of  the  fire  by  the  envelope  of  concrete. 
Concerning  this,  the  following  copy  of  a  testimonial  furnishes  evidence. 

TESTIMONIAL 

The  Wayss  &  Freytag  Co.,  are  hereby  informed  that  the  reinforced  concrete 
construction  carried  out  by  them  in  1901  in  my  factory  at  Neidenfels,  consist- 
ing of  floors,  beams  and  columns,  on  the  occasion  of  a  fire  on  the  5-6  of  Sept. 
prevented  the  spread  of  the  flames  to  the  factory  rooms  on  the  lower  stories. 

The  floor  was  found,  by  a  load  test  made  after  the  fire,  to  be  completely 


BUILDINGS 


213 


secure  at  all  points.  With  the  exception  of  the  finish  coat  on  the  floor,  which 
was  directly  exposed  to  the  fire  and  falhng  debris,  the  concrete  construction 
remained  absolutely  intact. 

I  can  therefore  recommend  the  above  mentioned  construction  to  all  whom 
it  may  concern,  as  absolutely  fire-proof. 

(Signed)       Julius  Glatz. 

Neidenfels,  Sept.  15,  1903. 
Rhine  Palatinate. 

The  foregoing  statement  of  the  firm  of  Julius  Glatz,  paper  manufacturer, 
Neindenfels,  describes  the  actual  circumstances.  The  load  test  of  a  part  of  the 
floor  in  question,  with  1800  kg.  to  the  sq.  meter  (369  lbs/ft^)  showed  that  the 
reinforced  concrete  construction,  with  the  exception  of  the  finish  coat,  had 
suffered  no  injury,  but  remained  entirely  intact. 

The  Royal  Fire  Insurance  Inspector. 

Neustadt-on-the-Hardt,  Sept.  16,  1903. 

Characteristic  occurrences  of  a  similar  nature,  are  recorded  in  Beton  und 
Eisen,"  Nos.  II  and  III,  1903.  The  superiority  of  the  concrete  protection  of  steel 
beams  and  columns,  as  a  means  of  security  against  fire,  compared  with  the 
terra  cotta  employed  in  America,  is  also  emphasized.  The  earthquake  and 
subsequent  fire  in  San  Francisco  left  unhurt  the  concrete  protection  of  beams 
and  columns,  while  the  terra  cotta  covering  fell  away  because  of  insufficient 
anchorage,  and  allowed  the  fire  free  access  to  the  steel  work.  Concerning 
entire  buildings  of  reinforced  concrete,  no  experience  can  be  gained,  since  the 
local  building  regulations  did  not  allow  the  erection  of  entire  structures  in  rein- 
forced concrete.  See  The  San  Francisco  Earthquake  and  Fire,  "  by  Himmel- 
wight;  also  "  Deutsche  Bauzeitung,"  1907,  No.  28. 

As  shown  in  the  accompanying  illustrations  (Figs.  206,  207)  of  a  spinning 
factory  in  Finland,  hangers  for  shafting,  etc.,  can  be  attached  at  any  point  to 
the  ceiling,  or  beams,  girders,  or  columns.  As  to  this  point,  preference  is  to  he 
given  to  a  ceiling  with  beams  spaced  2  to  3.5  m.  (6.5  to  11. 4  ft.)  apart,  compared 
with  large  paneled  ceilings  carried  directly  between  girders.  In  the  latter  case, 
special  stiffening  beams  must  be  provided  between  the  columns,  and  haunches 
between  the  floor  slabs  and  the  girders  are  also  to  be  recommended.  (See 
Fig.  207.) 

So-called  hanging  columns  may  also  be  advantageously  provided  for  holding 
shaft  hangers,  etc.  In  Fig.  208,  showing  the  interior  of  the  new  building  erected 
for  the  Speyer  cotton  mills,  some  of  these  members  are  illustrated.  Vibration, 
even  with  high  speed  machinery,  is  hardly  perceptible,  and  it  is  this  freedom 
from  susceptibility  to  shock  and  vibration  that  is  one  of  the  great  advantages 
of  reinforced  concrete  edifices.  As  a  rule,  the  elastic  deflection  of  a  reinforced 
concrete  beam  can  be  placed  at  ^  to  J  that  of  an  equally  large  steel  one.  With 
regard  to  vibration,  however,  the  great  weight  in  the  reinforced-concrete  floors 
and  beams,  rather  than  their  stiffness,  is  the  determining  factor. 

The  inclosing  of  the  supporting  structural  members  by  the  erection  of  fagade 
walls,  does  not  allow  of  the  utilization  to  the  fullest  extent  of  the  advantages  of 


214 


CONCRETE-STEEL  CONSTRUCTION 


BUILDINGS 


215 


reinforced  concrete.  They  are  much  better  reahzed  when  the  wall-beams  and 
columns  are  left  exposed  and  the  panels  thus  left  are  closed  with  brick  or  thin 
concrete  filling  walls.  In  such  cases,  walls  only  a  single  brick  in  thickness  on 
all  stories  will  suffice.  In  such  buildings,  the  wall  beams  must  Ije  made  heavy 
enough  to  carry  the  masonry  of  the  next  story  above.  This  consistent  utiliza- 
tion of  reinforced  concrete  work  is  as  stable  as  when  complete  inclosing  walls  of 
masonry  are  employed,  and  allows  a  material  saving  in  masonry  work  and 
foundations,  permitting  the  best  possible  use  to  be  made  of  the  site — an  important 
consideration  where  the  cost  of  land  is  as  great  as  in  cities.  In  buildings  erected 
without  a  cellar,  the  walls  can  be  carried  on  transverse  arches  or  reinforced 
concrete  girders,  extending  between  the  separate  foundations  of  the  wall  piers. 

An  example  of  this  type  of  construction  is  shown  in  Fig.  209,  and  a  complete 
lay-out  is  also  included,  together  with  details  of  the  reinforcement  for  the  new 
building  of  the  Daimler  motor  factory  in  Unterturkheim.* 

Fig.  210  shows  a  part  of  the  plan  of  the  ground  floor  and  Fig.  211  is  a  section 
of  the  building  which  has  a  length  of  131  m.  (429  ft.)  and  a  breadth  of  46  m.  (151 
ft.).  The  columns  throughout  the  building  are  spaced  5X5  meters  apart  (16.4 
ft.),  at  which  distance  are  also  spaced  the  girders  running  across  the  building. 
Perpendicular  to  them  are  the  beams,  with  a  spacing  of  2.5  m.  (8.2  ft.),  on 
which  the  floors  are  carried.  Over  the  room  on  the  front  of  the  building  are 
located  two  rows  of  girders  10  m.  (32.8  ft.)  long.  In  the  outer  walls  all  the  beams 
are  supported  on  reinforced  concrete  columns,  and  the  floor  panels  are  also 
there  carried  by  beams  of  the  same  material,  stretching  between  these  wall 
columns.  This  arrangement  creates  rectangular  panels  in  the  outside  walls,  which, 
because  of  the  great  width  of  the  building,  are  utilized  as  much  as  possible  for 
windows.  Except  the  brick  curtain  under  each  window,  with  concrete  window 
sills  and  a  small  brick  mullion  in  the  center  of  each  panel,  no  masonry  exists  in 
the  outside  waUs.  The  ribs  of  the  wall  beams  are  not  placed  beneath  the  floor 
slabs  as  is  customary,  but  are  immediately  above  them,  so  that  the  windows 
extend  up  to  the  under  side  of  the  floors.  (See  Fig.  214.)  The  second  floor 
is  provided  with  a  number  of  openings,  corresponding  with  the  roof  skylights, 
which  openings  are  covered  with  glass  and  contribute  to  the  better  illumination 
of  the  ground  floor.  The  concrete  wall  footings  extend  in  the  form  of  arches 
between  the  column  foundations.  In  the  front  of  the  building,  the  intermediate 
piers  are  also  executed  as  reinforced  concrete  columns,  Ijecause  in  that  case  they 
have  to  carry  the  loads  of  the  intermediate  beams. 

The  broken  lines  in  Fig.  210  of  the  plan  indicate  expansion  joints  which  traverse 
the  entire  structure,  including  the  floors  and  beams.  By  them  the  structure  is 
cut  in  two  longitudinally,  and  also  divided  into  Ave  parts  by  four  cross  joints. 
These  expansion  joints,  which  are  absolutely  necessary  in  large  buildings  to  pro- 
vide against  cracks  and  injurious  stresses,  were  introduced  for  the  first  time  in 
this  case,  as  far  as  known.  It  is  evident  the  arrangement  of  such  joints  does 
not  contribute  to  the  simplification  of  construction,  but  rather  complicates  the 
design.  The  joints,  which  were  constructed  closed,  later  opened  in  part  to  a 
width  of  6  mm.  (J  in.),  thus  affording  the  best  proof  of  their  necessity  and  of 

*  Deutsche  Bauzeitung,  Cement  Supplement,  No.  i,  1904. 


Fig.  209. — Office  building  of  Wayss  &  Freytag,  Neustodt-on-the-Hoardt. 


BUILDINGS 


217 


CONCRETE-STEEL  CONSTRUCTION 


Fig.  213. — Factory  of  the  Daimler  Motor  Co.  in  Unterturkheim  near  Stuttgart. 


BUILDINGS 


219 


their  practical  value.  Because  of  the  joint  running  longitudinally  through  the 
building,  the  auxiliary  beam  which  would  be  cut  by  it,  is  rc})laced  by  two 
smaller  ones,  over  which  the  floor  slab  is  cantilcvered  for  85  cm.  (33.5  in.).  Next 
the  elevators  the  expansion  joints  make  necessary  the  use  of  beam  brackets 
having  an  angle  less  than  45°,  and  in  the  roof  concrete,  ridges  with  zinc  coverings 
for  the  joints  had  to  be  provided,  as  shown  in  Fig.  211. 

The  roof  is  covered  with  pitch  strewn  with  gravel.  The  wall  beams  are 
constructed  above  the  roof  level,  thus  making  a  gravel  stop  unnecessary,  and 


Fig.  214. 


are  finished  with  a  molding  run  in  cement.  The  zinc  protecting  strip  engages 
in  a  joint  under  the  cornice.  From  Figs.  215  and  216  may  be  seen  the  details 
of  the  ground  floor  columns,  together  with  all  their  reinforcement.  The  columns 
under  the  girders  which  are  spaced  5  m.  (16.4  ft.)  apart,  have  a  section  of  32  X 
32  cm.  (13  in.),  with  a  reinforcement  of  four  round  rods,  20  mm.  (}  in.  approx.) 
in  diameter,  which  rest  at  the  bottom  on  a  flat  iron  grid,  and  at  intervals  of  20 
cm.  (7.9  in.)  are  connected  by  7  mm.  in.  approx.)  round  wire  ties.  The 
columns  under  the  girders  having  the  10  m.  (32.8  ft.)  spacing,  have  a  section  of 
40X40  cm.  (15.7  in.),  and  are  reinforced  with  four  20  mm.  (f  in.  approx.) 
round  rods  at  the  corners,  and  four  18  mm.  (^J  in.  approx.)  rods  between  them. 


220 


CONCRETE-STEEL  CONSTRUCTION 


The  section  of  the  wall  columns,  Fig.  217,  was  designed  with  regard  to  the 
window  openings.  The  reinforcement  consists  of  six  rods,  16  mm.  (f  in.)  in 
diameter. 


Figs.  215  and  216. — Ground  floor  interior  columns. 


The  live  load  for  the  second  story  was  600  kg/cm^  (123  Ib/ft^),  and  in  the 
computations  of  floors,  beams,  and  girders  the  most  unfavorable  distribution 
of  this  load  had  to  be  taken  into  consideration.  In  this  connection  it  was  assumed 
that  the  floor  slabs  would  rest  freely  on  the  beams,  that  these  would  be  supported 


BUILDINGS  221 

freely  on  the  girders,  and  that  the  latter  would  rest  freely  on  the  columns.  Thus, 
all  these  structural  meml)ers  would  be  continuous  beams  with  a  larger  or 
smaller  number  of  spans,  and  with  cantilever  ends  because  of  the  presence  of 
the  transverse  expansion  joints. 


*  ts  

H  W  j\ 


Fig.  217. — Ground  floor  wall  columns. 


In  Fig.  218  are  shown  the  positive  and  negative  moment  curves  for  a  girder 
of  four  spans  with  free  ends.  They  were  calculated  according  to  the  tabular 
values  given  by  Winkler,  and  the  influence  of  the  end  spans  is  only  considered 


222 


CONCRETE-STEEL  CONSTRUCTION 


with  reference  to  the  adjoining  intermediate  ones.  The  necessary  reinforcement 
at  top  and  bottom  was  detemined  on  the  basis  of  these  maximum  curves,  and 
was  constructed  as  shown  in  Fig.  219.  The  necessary  section  of  reinforcement 
at  the  top  over  the  supports,  was  provided  by  long  overlaps  of  the  bent  lower  bars. 


This  condition  cannot  be  obtained  from  the  small  overlaps  employed  in  certain 
systems."  In  the  computations,  the  restraint  of  the  beams  at  the  walls  was 
ignored,  but  since  a  part  of  the  lower  reinforcement  was  bent  upward  to  care  for 
the  shearing  forces,  a  partial  restraint  was  created  by  this  top  reinforcement  and 
its  anchorage  in  the  wall  columns. 

The  girders  join  the  columns,  and  the  beams  join  the  girders  with  haunches, 
so  that  the  allowable  compressive  stress  of  the  concrete  on  the  lower  surface  of 


BUILDINGS 


223 


the  beam  may  not  be  exceeded  at  those  points.  The  two  round  rods  i8  mm. 
(xg  in.  approx.)  in  diameter  which  pass  through  the  columns,  and  arc  carried 
into  the  girders  on  each  side,  serve  the  same  purpose.    The  reinforcement  of  th3 

beams  and  the  floor  slabs  was 
carried  out  on  the  same  principle. 
The  latter  is  shown  in  Fig.  220. 
In  Fig.  221  the  details  are  shown 
of  the  connection  of  the  two 
intermediate  beams  with  the  wall 
beams. 

The  25  cm.  (10  in.  approx.) 
brick  curtain  walls  originally 
planned  were  replaced  by  rein- 
forced concrete  ones,  8  cm. 
(3  in.  approx.)  thick,  which  were 
considered  equal  in  fire-proof 
quality  to  the  brick  curtains. 

The  construction  was  com- 
pleted in  three  months,  the  daily 
rate  for  floors  and  beams  being 
about  500  m^  (5381  ft^). 

This  structure  is  considered 
a  typical  example,  in  which  all 
the  advantages  of  concrete  con- 
struction were  secured,  so  that 
its  economic  benefits  were  also 
obtained.    The  short  period  of 


I  ^ 


3r 


Fig. 


erection  shows  that  with  the  necessary  trained  staff  and  the  required  equipment, 
reinforced  concrete  buildings  may  be  erected  with  a  speed  of  which  no  other 
type  of  building  admits. 


224 


CONCRETE-STEEL  CONSTRUCTION 


Figs.  222  and  223  show  examples  of  the  tasteful  interior  treatment  of  the  Tietz 
Shop  on  the  Bahnofplatz  in  Munich  (Heilmann  and  Littmann,  architects).  The 
same  rectangular  reinforced  concrete  panel  was  employed  on  all  fleers.  The 
doubly  reinforced  slabs  with  spans  of  5.15  m.  (16.9  ft.)  in  both  directions  had 
haunches  on  all  four  sides  at  the  beams,  which  were  all  of  equal  size.  In  the 
center  of  the  building  rose  a  large  light  shaft  of  elliptical  section.  The  illustra- 
tions show  the  completed  decorative  interior  in  reinforced  concrete,  the  light 
shaft  with  stairways  and  the  counter  space.  An  extremely  simple  architectural 
treatment  of  the  interior  of  the  building  was  intended.    The  entire  under  sides 


Fig.  222. — Interior  view  of  the  Tietz  shop  in  Munich. 


of  the  floors  and  beam  bottoms  with  their  light  moldings  were  made  white, 
down  to  the  two  brown  lines  around  the  tops  of  the  columns,  making  a  very 
simple  decorative  treatment  of  the  interior  of  the  salesrooms.  The  reinforced 
concrete  columns  and  parapets  around  the  light  well  were  covered  with  tasteful 
marble  panels  in  different  colors,  and  decorated  with  colored  mosaic  glass  tile. 
Fig.  224  shows  the  placing  of  the  reinforcement. 

The  double  reinforcement  of  the  floor  slabs  was  done  with  a  beam  spacing 
of  6.5X6.5  m.  (21.3  ft.).  It  is  possible,  however,  to  divide  the  panels  produced 
by  a  square  column  spacing  into  smaller  squares  by  beams  crossing  each  other, 
reinforcing  the  slabs  as  square  panels  partially  restrained  at  the  supports.  Such 
an  arrangement  is  shown  in  Fig.  225.  The  beams  then  rest  freely  on  the  inclos- 
ing walls. 


BUILDINGS 


225 


Fig.  224 — Placing  reinforcement  for  Tietz  shop. 


226 


CONCRETE-STEEL  CONSTRUCTION 


Fig.  227  shows  the  floors  of  the  immense  Deimhardt  wine  cellar  during  con- 
struction in  Coblenz.    Since  only  a  small  height  could  be  employed  for  the 


Fig.  225. — Storehouse  in  Ulm.    Square  floor  panels  and  intersecting  floor  beams. 


structure,  and  at  the  same  time  large  spans  and  considerable  live  loads  (1000 
and  1500  kg/m^ — 200  and  300  lbs/ft^)  were  required,  intermediate  beams  were 
necessarily  omitted  and  the  floor  panels  were  given  a  width  of  6.1  m.  (20  ft.)  between. 


Fig.  226. — Printing  house  in  Heibronn. 


girders.  Because  of  the  small  head  room,  it  was  necessary  to  make  the  latter 
very  wide. 

The  floors  were  constructed  according  to  the  ZoUner  cellular  system,  in  which 
gutters  were  placed  throughout  the  central  parts  of  the  panels. 


BUILDINGS 


227 


Fig.  227. — Erection  of  the  three-story  Deinhardt  wine  cellar,  Coblenz. 


Fig.  228. — Cross-section  of  the  Singer  Manufacturing  Co.  factory  in  Wittenberg, 


228 


CONCRETE-STEEL  CONSTRUCTION 


There  may  also  be  mentioned  the  immense  factory  now  in  course  of  construc- 
tion for  the  Singer  Manufacturing  Co.,  in  Wittenberg,  near  Hamburg.    It  covers 


Fig.  229. — View  of  a  floor  in  the  Singer  Manufacturing  Co.  factory  showing  widening  of  the 
beams  where  they  intersect  the  girders.    Columns  of  beton  frette. 


Fig,  230 — Krefeld  warehouse,  with  flat  roof  and  monitor  skylights. 

an  area  of  over  5000  m^  (54,000  ft^  approx.)  while  the  whole  building  with  its 
four  floors  affords  a  working  space  of  about  20,000  m^  (240,000  ft^  approx.).  Teh 


BUILDINGS 


229 


Fig.  233. — Loft  of  the  forge  shop  of  the  Daimler  motor  factory. 


230 


CONCRETE-STEEL  CONSTRUCTION 


floors,  which  had  to  carry  a  heavy  hve  load,  could  have  only  shallow  intermediate 
beams,  because  of  the  power  transmission  requirements,  and  for  the  same  reasons 
they  could  not  be  very  deep  at  the 

girders.       The     heavy     compressive  . 


Fig.  234. — Hall  in  Pfersee.  Fig.  235 — Locomotive  house  in  Haben  Krefeld. 


The  roof  covering  best  adapted  for  reinforced  concrete  is  of  tar,  laid  with  a 
grade  of  2j%.    The  10  cm.  (4  in.  approx.)  layer  of  gravel  on  such  a  roof,  in  most 


Fig.  236. — Interior  veiw  of  locomotive  house  in  Krefeld. 


BUILDINGS 


231 


cases  provides  sufficient  insulation  against  changes  of  temperature.  A  roof  cover- 
ing with  a  double  layer  of  felt  is  also  employed,  and  has  the  advantage  of  lower 
cost,  and  allows  a  steeper  pitch,  up  to  about  15%.  Sheet  zinc  is  attacked  by 
the  cement  concrete,  and  requires  an  intermediate  layer  of  rooting  ])a])er.  In 
buildings  of  large  area,  some  roof  panels  may  be  omitted,  and  subsecjuently  covered 
with  monitor  skylights.  (See  Fig.  230.)  For  architectural  or  other  reasons, 
high  pitched  roofs  are  also  constructed.  They  may  be  covered  with  sheet  metal 
or  tile.  (See  Fig.  231.)  Roofs  are  also  constructed  with  long  span  girders. 
Fig.  234  shows  the  roof  of  a  hall  in  Pfersee,  with  girders  in  the  form  of  l)raced 
arches,  while  Figs.  235-237  show  a  locomotive   roundhouse  at  Hafen  Krefeld, 


Fig.  237. — Exterior  view  of  locomotive  house  in  Krefeld. 


in  which  the  girders  are  in  the  form  of  arched  beams.  The  design  of  the  latter 
can  be  done  through  consideration  of  the  elastic  conditions  produced  by  live  load, 
and  snow  and  wind  pressures. 

The  forms  for  the  floors  of  reinforced  concrete  buildings  several  stories  in 
height,  are  best  so  arranged  that  the  side  pieces  of  the  beams  form  can  be  removed 
after  the  concrete  has  hardened  sufficiently,  w^hereas  the  supports  and  bottom 
pieces  should  be  allowed  to  remain  for  a  longer  period  (4  to  6  weeks).  To 
facilitate  the  removal  of  the  forms,  all  the  supports  rest  on  wedges.  In  determin- 
ing the  period  for  removing  the  forms,  besides  the  weather  conditions  during 
the  period  of  setting,  the  question  has  to  be  considered  as  to  whether  the  floor 
in  question  has  to  carry  the  forms  for  the  stories  above  it. 

Spirally  reinforced  concrete  is  especially  applicable  to  the  columns  of  high 


232  CONCRETE-STEEL  CONSTRUCTION 

buildings.  Above  all,  its  usefulness  is  disclosed  where  heavily  loaded  columns 
would  require  large  diameters,  such  as  those  shown  in  Fig.  229  in  the  new  build- 
ing for  the  Singer  Co.,  which  are  supplied  with 
spirals.  They  were  wound  on  hollow  cylinders, 
with  the  coils  close  together,  and  could  then  be 
easily  stretched  longitudinally  until  the  required 
pitch  was  secured.  The  last  half- turn  of  the 
spiral  should  overlap,  and  the  end  of  the  rod  be 
made  into  a  hook  carried  to  the  center  of  the 
coil.  Fig.  238  shows  the  steel  skeleton  of  such 
a  column. 

STAIRS 

Reinforced  concrete  stairs  may  be  variously 
applied.  In  Fig.  239  is  shown  a  staircase  built 
according  to  the  old  method  after  the  Monier 
system;  the  flights  being  supported  by  flat  arches, 
braced  against  the  structural  parts  of  the  land- 
ings. The  steps  are  of  concrete.  Winding 
staircases  in  residences  (Fig.  240)  may  also  be 
erected  in  reinforced  concrete,  by  arranging  as 
the  carrying  structure,  a  winding  slab  of  rein- 
forced concrete  from  10  to  14  cm.  (4  to  6  in. 
approx.)  thick,  supported  in  slots,  6  to  10  cm.  (2J 
to  4  in.  approx.)  deep,  cut  in  the  masonry,  on 
which  slab  the  steps  are  formed.  Winding 
stairs   with  side  strings  are  also   practical,  as 

FlG.238.-Steel  skeleton  of  a  spirally  shown  in  Fig.  241. 

reinforced  column.  Stairs  with  Straight  flights  can  be  arranged 

in  different  ways,  according  to  conditions.  Thus, 
Fig.  242  shows  a  stairway  of  reinforced  concrete,  in  which,  for  lack  of  other 
support,  the  landing  is  suspended  from  the  reinforced  concrete  beam  overhead; 
while,  in  Fig.  243,  an  arrangement  is  exhibited  in  which  brackets  of  reinforced 
concrete  are  employed,  which  bisect  the  angle  of  the  inclosing  walls,  each  forming 
an  integral  part  of  one  tread,  thus  carrying  the  flights. 

Artificial  stone  stairs  can  also  be  constructed  of  concrete,  each  separate  step 
forming  a  reinforced  concrete  beam,  molded  in  advance,  and  set  in  place  after 
having  sufficient  time  to  harden.  Both  ends  may  be  supported,  or  in  the  case  of 
flying  staircases,  one  end  may  be  set  in  a  wall. 

Properly  constructed  stairways  of  reinforced  concrete  are  just  as  safe  as 
other  parts.  This  cannot  always  be  said  of  stairways  built  of  stone  or  wood,  in 
accordance  with  usual  methods.  In  regard  to  security  against  fire,  reinforced 
concrete  stairs  are  superior  not  only  to  those  of  wood  or  stone,  but  also  to  those 
of  iron.  The  little  dependence  which  can  be  placed  on  stairs  built  of  limestone 
is  shown  in  an  illustration  in  ''Beton  und  Eisen,"  No.  II,  1903,  p.  79. 


STAIRS 


233 


CONCRETE-STEEL  CONSTRUCTION 


234 


The  treads  of  reinforced  concrete  stairs  can  be  finished  with  any  desired 
covering,  such  as  linoleum,  oak,  etc.,  so  that  they  are  adapted  to  the  highest 
requirements. 


Fig   243  — Reinforced  concrete  stairs  in  the  storehouse  of  the  National  Railway,  Elberfeld  in 

Opladen. 


ARCHES  IN  BUILDINGS 


235 


ARCHES  IN  BUILDINGS 

The  archfcs  which  occur  in  buildings  may  serve  the  most  varied  ends.  If  they 
are  merely  decorative  in  character,  they  may  be  executed  as  thin  ''Monier" 
arches,  without  forms,  with  woven  wire  on  an  iron  framework.  They  are  then 
more  easily  carried  and  more  durable  than  when  built  of  "  Rabitz  "*  construction. 

By  employing  heavier  T  and  I>  iron  framing,  larger  openings  may  be  spanned 
without  forms.  This  method  has  been  employed  for  numerous  vaulted  church 
roofs,  as  shown  in  Fig.  245.  The  framing  iron  is  there  located  within  the  arch 
ribs. 

In  distinction  from  this  may  be  mentioned  the  work  shown  in  Fig.  246,  of 
the  vaulted  ceiling  of  St.  Joseph's  Church  in  Wurzburg,  for  which  the  arch  ribs, 
with  a  span  of  20  m  (65.6  ft.)  were  rammed  in  wooden  forms  and  reinforced 
with  round  rods. 

In  order  to  give  the  ribs  the  same  appearance  as  the  work  below  the  spring 
of  the  arches,  which  was  composed  of  genui::e  shell  limestone,  crushed  stone, 
and  grit  for  the  concrete,  was  prepared  from  the  same  material,  and  the  ribs 
afterwards  dressed  by  a  stone  cutter.  The  appearance  of  these  arches  fully  met 
all  requirements.  The  inverts  between  the  ribs  were  made  after  the  Holzer 
patents,  small  framing  irons  being  employed  :ind  giving  excellent  results.  The 
advantages  of  this  type  of  arch  are  its  security  against  cracking  and  its  stability 
compared  with  a  stone  structure.  In  case  of  fire,  such  a  vault  will  withstand  the 
fall  of  the  burned  roof  with  perfect  safety,  and  thus  it  forms  an  important  pro- 
tction  for  the  interior  of  the  church.  Manifestly  all  such  barrel  or  groined 
arches  can  be  constructed  wholly  on  forms. 

Fig.  247  shows  a  fire-proof  arrangement  of  a  continuous  barrel  arch  with 
transverse  openings  over  the  gymnasium  of  a  school  house  in  Munich. 

Among  other  uses  of  the  reinforced  concrete  arch  in  building  work  may  be 
mentioned  an  arch  in  which  the  profile  which  is  required  for  architectural  reasons, 
does  not  conform  to  that  necessary  from  a  statical  point  of  view.  Only  in  rein- 
forced concrete  can  such  structures  be  executed — an  example  being  shown  in  Fig. 
249 — since  tensile  stresses  can  then  be  cared  for  and  the  line  of  pressure  need  not 
necessarily  be  located  within  the  middle  third. 

Reinforced  concrete  is  particularly  well  adapted  for  the  construction  of  domes. 
Through  variation  in  the  number  and  size  of  the  reinforcing  members  placed  in 
meridinal  and  parallel  circles,  it  is  possible  to  provide  for  all  possible  stresses. 
Since  reinforced  concrete  will  resist  tensile  stresses,  almost  any  surface  of  revolu- 
tion can  be  used  as  the  dome.  Around  its  l^ase  a  tension  ring,  preferably  of 
channel  section,  resists  all  the  horiontal  components  of  the  meridinal  stresses, 
so  that  the  interior  is  entirely  free  from  structural  parts.  If  the  crown  of  the  dome 
is  to  be  broken,  so  as  to  admit  a  skylight  or  cupola,  the  meridinal  compressive 
stresses  can  be  resisted  by  a  concrete  pressure  ring  with  reinforcement. 

Domes  (6  to  12  cm.  thick — 2.4  to  4.7  in.)  can  be  erected  with  the  help  of  forms 


*  Cement  mortar  on  wire  lath. — (Trans.) 


CONCRETE  STEEL  CONSTRUCTION 


Fig.  245. — Groined  arches  of  reinforced  concrete  in  the  monastery  chapel  in  Landau. 


ARCHES  JN  BUILDINGS 


237 


Fig.  246. — Arches  in  St.  Joseph's  Church  in  Wiirzburg. 


■Fig.  247. — Fireproof  Monier  arches  over  the  gynasium  of  the  school  in  the  Gatzingerplatz, 

Miinich. 


238 


CONCRETE-STEEL  CONSTRUCTION 


Fig.  249 — Reinforced  concrete  semi-circular  arches  in  the  cupola  of  the  new  station  in.  Niirnberg. 


239 


or  with  shaping  members  in  the  directions  of  the  meridinal  and  parallel  circles, 
the  trapezoidal  spaces  between  irons  being  covered  with  woven  wire,  held  in 
place  by  wires  run  through  holes  in  the  frames.    The  concrete  is  then  made  to 


Fig,  250 — Reinforced  concrete  cupola  and  lantern  of  the  Army  Museum  in  Munich,  diameter 

36  m.  (52.5  ft.). 

cover  the  whole  of  the  metal.  The  design  of  domes  is  best  done  graphically, 
just  as  in  domes  of  the  Schwedler  type. 

Figs.  250-253  show  a  reinforced  concrete  dome  of  the  Army  Museum  in 
Munich. 

In  Fig.  250  is  shown  the  exterior  view  of  the  whole  dome.    Including  the 


240 


CONCRETE-STEEL  CONSTRUCTION 


9  m.  (30  ft.)  high  lantern 
(187  ft.). 


ith  its  spire,  the  total  height  above  grade  is  57  m. 


Fig.  253  is  a  plan  of  the  dome,  and  in  Fig.  252  a  section  is  shown.  Fig. 


251  shows,  at  a  larger  scale,  the  support  for  the  base  of  the  dome 
shown,  both  inner  and  outer  shells  were  provided.  The 
first,  of  a  radius  of  8.1  m.  (26.6  ft.),  carries  only  its  own 
w^eight  and  is  supported  about  i  m.  (3.3  ft.)  lower  than 
the  outer  dome,  the  surface  of  which  is  generated  by  radii 
of  varying  size.  Each  dome  is  supported  on  a  footing 
ring,  consisting  of  a  D.N.P.*  14  cm.  (5.5  in.)  channel, 
set  in  the  tambour,  which  had  a  thickness  of  from  38 
to  51  cm.  (15  to  20  in.).  The  concrete  was  from  5  to  6 
cm.  (2  to  2.4  in.)  thick,  and  was  strengthened  by 
reinforcement  arranged  in  the  directions  of  the  meridinal 
and  parallel  circles.  In  the  outer  dome  the  upper  com- 
pression ring  is  made  of  an  angle  50X50X7  mm. 
(2X2X1^  in.  approx.),  the  ring  upon  which  the  lantern 
4  m.  (13. 1  ft.)  in  diameter  rests,  is  a  D.N.P.  12X6  cm. 
T  (4.7X2.4  in.),  while  the  remaining  parallel  circles 
are  8x4  cm.  T's,  (3.iXi.6in.)  and  the  meridians 


As  there 


Fig.  251. — Detail  of  base 
of  cupola. 


Fig.  252. — Vertical  section  through  the  cupola  of 
the  Army  Museum  in  Munich. 


are  9x4.5  cm.  T's  (3.5X1.8  in.).  Between  the  meridinal  and  ring  members 
was  placed  a  7  mm.  (3^  in.  approx.)  round  wire  grid,  with  10  cm.  (3.9  in.) 
meshes.    The  inner  dome  is  covered  only  with  mortar  on  the  outer  side,  the 


German  standard. — (Trans.) 


SPREAD  FOOTINGS 


241 


outer  dome  being  covered  with  copper,  fastened  to  wooden  plugs  which  were 
placed  in  the  dome  along  with  the  concrete.  The  copper  is  separated  from  the 
concrete  by  a  coat  of  asphalt. 


0                                                       5  10*" 
 1  1  ^  i  1  i  J  4  ^ 

Fig.  253. — Plan  of  cupola  of  the  Army  Museum. 


SPREAD  FOOTINGS 

To  prevent  bad  settlements,  the  areas  of  the  foundations  of  a  bulding  must 
be  proportioned  to  the  carrying  capacity  of  the  soil.  If  the  carrying  capacity 
is  small,  a  very  large  bearing  area  is  required,  which  can  be  secured  by  the  use 
of  reinforced  slabs  possessing  the  necessary  bending  strength.  In  this  way  it 
is  possible  to  reduce  the  pressure  on  the  soil  to  0.5  kg/cm^  (0.5  tons/ft^)  or  less, 
so  that  poor  building  areas  and  filled  ground  can  be  used  for  the  foundations. 

Fig.  254  shows  the  bottom  of  a  sewer  in  Wiesbaden,  with  footings  8  m.  (26  ft.) 
deep.  Since  the  soil  could  not  be  loaded  more  than  1.5  kg/cm^  (1.5  tons/ft^), 
the  entire  space  between  walls  had  to  be  called  upon  to  resist  the  pressure.  This 
was  accomplished  by  the  construction  of  a  foundation  slab,  45  cm.  (17.7  in.) 
thick,  extending  from  one  wall  to  the  other.  Considered  statically,  the  slab 
constituted  a  beam  supported  at  both  ends,  and  bent  by  the  upward  earth  pressure. 
The  reinforcement  required  per  meter  length  of  sewer  was  10  rods  24  mm.  (i^^  in.) 
in  diameter,  which,  because  of  the  curvature  of  the  surface,  had  to  be  anchored 
into  the  concrete  by  means  of  stirrups  7  mm.  (yq  in.  approx.)  thick.  As  a  sub- 
base  for  the  sewer,  a  15  cm.  (6  in.  approx.)  layer  of  stone  was  laid  in  cement 
mortar. 

Spread  footings  should  be  employed  when,  for  any  reason,  piling  is  imprac- 
ticable.   In  many  instances,  such  footings  are  cheaper  than  piling. 

Fig.  255  shows  a  usual  form  of  reinforced  concrete  footing,  under  a  row  of 
silo  columns.  In  this  instance  it  was  necessary  to  distribute  the  concentrated 
column  load  uniformly  longitudinally  as  well  as  laterally,  and  consequently 


CONCRETE  STEEL  CONSTRUCTION 


242 


reinforcement  was  needed  in  both  directions.  The  longitudinal  reinforcement 
served  to  distribute  the  concentrated  loads  over  the  space  beween  the  columns. 


Fig.  254. — Reinforced  concrete  sewer  base  in  Wiesbaden, 


Walls  of  buildings  can  be  founded  on  reinforced  concrete  footings.  For  this 
purpose  slabs  of  necessary  width  should  be  constructed  along  the  line  of  the 
wall,  projecting  equally  on  both  sides,  as  shown  in  Fig.  255.    Such  footings  do  not, 


Fig.  255. — Silo  column  foundation. 


however,  distribute  the  load  uniformly  if  eccentrically  located  with  respect  to  the 
wall,  as  sometimes  happens  when  the  wall  is  built  flush  with  the  property  line.  In 
such  a  case  a  slab  is  required  which  extends  under  the  whole  building,  or  a 


SPREAD  FOOTINGS 


243 


244 


CONCRETE-STEEL  CONSTRUCTION 


large  part  of  it.  If  the  centroid  of  the  slab  coincides  with  that  of  the  building 
load,  it  is  possible  to  distribute  the  load  uniformly  over  the  site,  by  means  of  a 
reinforced  concrete  foundation. 

In  Fig.  256  is  shown  the  upper  and  lower  reinforcement  of  a  footing  slab,  50 
cm.  (20  in.)  thick  under  a  business  block  in  Stuttgart.  Upon  the  assumption 
of  a  uniform  earth  pressure  of  0.7  kg/cm^  (0.7  tons/ft^)  it  was  possible  to  compute 
approximately  the  positive  and  negative  moments  of  the  various  panels  and  to 
arrange  the  reinforcement  accordingly.  Between  the  piers  a  stronger  reinforce- 
ment is  provided,  so  that  the  slab  will  act  as  a  beam  to  distribute  the  concentrated 
load  uniformly  over  its  entire  length,  and  transmit  it  to  the  line  of  reinforcement 
running  perpendicular  to  the  heavier  material.  At  distances  of  about  50X50 
cm.  (20X20  in.  approx.)  the  upper  and  lower  reinforcement  was  tied  together 
with  stirrups.  Similar  footings  as  much  as  75  cm.  (30  in.  approx.)  thick  were 
constructed  by  Wayss  and  Freytag  for  the  large  Elbhof  block  in  Hamburg,  and 
for  various  large  silos. 

As  a  rule,  in  complicated  ground  plans,  the  statical  relations  are  not  entirely 
clear,  so  that  calculations  are  made  on  somewhat  unfavorable  assumptions,  and 
a  certain  excess  of  reinforcement  should  be  employed.  Sometimes  the  computa- 
tions cannot  be  based  on  a  uniform  soil  pressure.  In  such  cases  the  resultant  of 
all  the  loads  does  not  coincide  with  the  centroid  of  the  ground  plan,  or  some  of  the 
loads  are  much  heavier  than  others.  Then  varying  pressures  are  to  be  reckoned 
under  the  several  parts. 

Foundation  slabs  constructed  in  this  manner  permit  of  a  saving  of  excavation, 
require  less  material,  and  solve  the  problem  better  than  when  the  usual  heavy 
concrete  footings  are  employed,  reinforced  with  grillages  of  rails  or  I-beams.  They 
are  adapted  to  the  foundations  of  buildings,  chimneys,  fountains  with  heavy  super- 
structures, monuments,  etc.,  which  are  to  be  set  on  light  soils. 

SUNKEN  WELL  CASINGS 

Sunken  well  casings  of  reinforced  concrete  were  constructed  according  to  the 
Monier  system  at  an  early  period  of  its  development.  They  serve  either  for 
obtaining  water,  in  which  case  they  are  provided  with  holes  in  the  walls,  or  as 
foundations  for  buildings,  bridge  piers,  etc.,  and  must  be  filled  with  concrete  after 
being  sunk.  Compared  with  masonry  well  casings,  they  are  capable  of  resist- 
ing heavier  external  pressures,  and  on  account  of  the  thinness  of  their  walls  they 
easily  penetrate  the  soil,  but  their  relatively  lighter  weight  makes  necessary  a 
greater  artificial  load. 

Fig.  257  shows  such  a  well  being  sunk  by  hand  dredging  for  the  municipal 
plant  of  Bamberg.    The  top  is  constructed  in  the  shape  of  a  dome. 

In  Fig.  258  are  shown  the  piers  for  the  Kocher  Bridge  at  Brockingen,  erected 
on  such  sunken  wells,  details  of  reinforcement  being  also  given.  The  casings, 
with  a  clear  width  of  1.5  m.  (5  ft.  approx.)  or  less,  were  constructed  as  reinforced 
concrete  pipes,  those  of  the  larger  diameter  being  constructed  in  situ  between, 
forms.  Over  the  well  is  laid  a  heavy  slab  of  reinforced  concrete,  upon  which  the 
masonry  rests. 


SUNKEN  WELL  CASINGS 


245 


An  extensive  water  supply  system  was  built  in  '1902  for  the  Pasinger  Pai)er 
Mill  (see  Fig.'  259).    The  method  of  construction  was  as  follows:     First,  a 


Fig.  257. — Wells  for  the  municipal  electric  plant  of  Bamberg. 


I 


Fig.  258. — Well  casing  foundations  for  the  Kocher  Bridge  at  Brockingcn. 

circular  excavation  was  made  in  the  gravelly  soil,  with  fairly  steep  banks  down  to 
elevation  0.0.    Then  the  [  ]  rings  of  the  upper  part  of  the  well,  fastened  together 


Fig.  259. — Reinforced  concrete  well  for  the  Pasingen  paper  mill. 


WATER-TIGHT  CELLARS 


247 


vertically  by  round  rods,  were  erected,  commencing  at  the  bottom,  and  outside 
them  wooden  planks  were  j)laced  and  driven  down  as  the  excavation  was  dee{)ened. 
At  the  same  time  the  rings  below  the  o.o  level  were  placed,  so  that  between  grades 
o.o  and  — 3.3  m.,  the  shaft  was  excavated  under  partially  water-tight  conditions 
with  vertical  sides.  At  this  j)oint  the  construction  of  the  casing  proj)er  was 
commenced  at  level  1.5,*  this  being  above  water  level.  Because  of  the  heavy 
compressive  stresses  to  which  the  casing  would  l)e  subjected,  it  was  constructed 
of  reinforced  concrete,  with  inwardly  j^rojecting  ribs  reinforced  with  es])ecially 
heavy  channels  and  round  rods.  The  outside  surface  was  finished  smooth,  and 
the  lower  edge  was  provided  with  a  cutting  ring.  During  the  sinking  oj)erati()n, 
sub-aqueous  excavation  was  not  employed  because  of  practical  considerations, 
the  water  being  kept  down  and  the  material  removed  by  mechanical  api)liances. 
To  drain  the  excavation,  large  centrifugal  ])umps  and  electric  motors  were  set  up 
on  steel  elevators  erected  on  the  heavy  ribs  of  the  casing,  so  that  the  machinery 
could  be  easily  raised,  should  the  electric  current  fail.  Upon  the  completion 
of  the  sinking  of  the  casing,  the  concreting  of  the  lining  and  of  the  reinforcing  rings 
above  level  — 3.3  was  undertaken,  and  at  the  same  time  a  water-tight  reinforced 
concrete  floor  was  installed  over  about  |  of  the  area  of  the  well,  and  a  reinforced 
concrete  wall  carried  from  this  floor  above  the  level  of  the  highest  ground  water. 
In  this  way  a  convenient  entrance  was  provided  to  the  well  proper,  as  well  as 
a  perfectly  water-tight  spacious  pump  chamber,  at  a  depth  which  lessened  the 
suction  on  the  pumps.  Iron  steps  made  pump  pit  and  well  easily  accessible. 
Above  ground,  the  well  was  inclosed  by  a  small  building  which  accommodates 
the  power  machinery. 

Concrete  and  reinforced  concrete  pneumatic  caissons  are  discussed  in  a  paper 
by  K.  E.  Hilgard,  in  the  Transactions  of  the  American  Society  of  Civil  Engineers, 
Vol.  63,  1904.  According  to  this  paper,  such  caissons  have  been  used  in  large 
numbers  in  Switzerland  for  foundations  for  bridge  piers  in  the  correction  of 
tlie  course  of  the  Rhine,  and  as  foundations  of  turbine  pits. 


WATER=TIQHT  CELLARS 

In  all  cases  where  the  highest  ground  water  level  is  above  the  bottom  of  a 
cellar  which  it  is  desired  to  maintain  in  a  useful  condition,  a  water-proofing  of 
the  walls  and  bottom  is  necessary.  An  excellent  method  of  water-proofing  the 
floor  is  to  employ  shallow  inverted  reinforced  concrete  arches,  which  span  directly 
between  the  foundation  walls  in  the  form  of  cyhndrical  arches,  or  are  built  as 
groined  arches  between  the  walls  and  the  intermediate  column  foundations  of 
the  building.  On  these  arches  a  water-tight  cement  finish  is  laid  which  really 
acts  as  the  water-excluding  medium,  and  consequently  must  be  protected  from 
injury  by  a  concrete  filHng  over  it.  This  filling  concrete  is  leveled  and  finished 
with  a  wearing  coat.  At  the  walls,  the  arches  are  anchored  and  continued  as 
a  v/all  coating  of  concrete  with  proper  reinforcement,  on  which  concrete  the 


*  Probably  -  1.5.— (Trans.) 


248 


CONCRETE-STEEL  CONSTRUCTION 


water-proofing  is  applied.  Should  the  purpose  for  which  the  cellar  is  to  be  used 
so  demand,  the  wall  coat  must  also  be  protected  from  injury. 

In  Fig.  260  is  shown  the  woven  wire  reinforcement  in  a  portion  of  the  new 
building  for  the  Daimler  Motor  Company.  The  cellar  construction  was  first 
planned,  so  that  the  column  foundations  were  to  be  walled  and  coated.  A 


Fig.  260 — Woven  wire  reinforcement  for  a  water-tight  cellar  in  reconstruction  of  the  Daimler 

motor  factory  in  Unterturkheim. 

certain  amount  of  continuity  of  construction  beneath  the  foundations  was  effected 
by  pumping  cement  grout  into  the  gravelly  soil  immediately  under  the  columns. 
The  arches  are  inverted  groined  ones.  In  Fig.  260,  the  reinforced  coating  of  water- 
proof cement  for  the  column  foundation  is  shown  completed,  while  the  metal 
grid  for  the  floor  arches  is  clearly  visible. 


Fig.  261. — Reinforcement  of  the  beams  of  a  water-tight  cellar. 

Another  form  of  construction  with  floor  arches  8  m.  (26  ft.)  wide  abutting 
against  reinforced  concrete  beams,  is  shown  in  Fig.  262.  The  necessity  of 
water-proofing  was  discovered  only  at  a  late  stage  of  construction,  and  groined 
arches  were  impracticable,  because  of  the  long  rectangular  spaces  between 
columns.  Wedge-shaped  reinforced  concrete  beams  were  therefore  constructed 
between  the  column  foundations,  and  between  the  beams,  flat,  8  m.  (26  ft.)  span, 


WATER-TIGHT  CELLARS 


249 


barrel  arches  were  built.  In  Fig.  261  the  details  of  the  reinforcement  are  shown, 
especially  the  arrangement  of  the  beam  reinforcement  to  withstand  shearing 
stresses.    The  ground  water  level  was  1.3  m.  (4.2  ft.)  above  the  cellar  bottom. 


Fig.  262. — Section  through  beams  and  arches  of  a  water-tight  cellar. 

Similar  proofing  against  ground  water  is  often  required  in  the  boiler  pits  of 
steam  heating  systems.  In  this  case,  however,  the  water-proof  cement  coating 
must  be  made  to  resist  the  radiant  heat  of  the  boilers. 


Fig.  263. — Pump  house  of  the  Wolsum  pulp  mil 


In  the  pump-house  of  the  pulp  mill  at  Walsum,  the  bottom  of  the  pump 
chamber  had  to  be  water-proofed  against  a  high  water  level,  7  m.  (23  ft.)  above 


250 


CONCRETE-STEEL  CONSTRUCTION 


the  floor.  Because  of  this  heavy  pressure  two  reinforced  floor  arches,  one  over 
the  other,  were  employed,  each  with  a  water-proof  layer,  and  both  joined  to 
the  reinforced  waU  covering.     The  arrangement  is  shown  in  section  in  Fig.  263. 


PILES 


Spiralen 
^10""" 


Reinforced  concrete  bearing  piles  and  sheet  piles  have  the  advantage  when 
compared  with  those  of  wood,  that  the  concrete  ones  may  be  used  above  ground 
water  level,  whereby  a  considerable  saving  on  the 
whole  foundation  may  be  effected.  The  reinforcing 
of  the  common  square  pile  corresponds  exactly  with 
that  of  a  reinforced  column,  the  longitudinal  rods 
being  combined  at  the  pointed  end  in  an  iron  shoe. 
A  good  connection  between  the  longitudinal  rods  by 
ties  spaced  not  too  far  apart  is  very  important,  just 
as  for  columns.  Further,  so  as  to  distribute  the  impact 
during  driving,  it  is  necessary  to  interpose  some  medium 
between  the  pile  driver  hammer  and  the  pile,  and  also 
use  a  solid  cap  inclosing  the  head  of  the  pile,  to 
prevent  its  destruction.  This  variety  of  pile  was  intro- 
duced in  excavation  work  by  Hennebique.  A  detailed 
description  of  the  subject  was  given  by  Diemling  in 
"  Beton  und  Eisen,  "  No.  II,  1904. 

On  account  of  its  high  compressive  strength,  spirally 
reinforced  concrete  is  especially  suitable  for  piles.  The 
section  is  then  usually  circular  or  octagonal.  Because 
of  the  excellent  results  which  have  been  attained  in 
Paris  on  Considere  piles,  which  have  been  driven  without 
any  protecting  device,*  Wayss  and  Freytag  acquired 
the  exclusive  rights  to  spirally  reinforced  piles  and 
have  secured  excellent  results  in  numerous  instances. 
Fig.  264  shows  the  construction  of  one  of  the  reinforced 
piles  for  the  top  of  the  600  m.  (2000  ft.  approx.)  long 
and  37  m.  (121  ft.)  wide,  Ill-Hoch  canal  in  Miilhausen 
in  Alsace.  The  driving  took  place  when  the  con- 
crete was  only  six  weeks  old.  The  load  to  be  carried 
by  each  pile  was  36  t.  (39.6  tons);  the  reinforcement 
consisted  of  8  longitudinal  rods  14  mm.  in.)  in 
diameter  and  a  spiral  of  10  mm.  (J  in.)  material 
Fig  264  —  Reinforced  ^  pitch  of  6  cm.  (2.4  in.)  which  was  reduced  to  3 

concrete  pile  for  the         (^'^  head  and  the  point.    The  wrought- 

top  of  the  canal  in   iron  point  had  four  prongs,  the  upper  ends  of  which 
Mulhausen  in  Alsace,   were  slightly  offset  and   were  punched.     At  the  offset 
was  a  turn  of  the  spiral  and    the    holes  provided  a 
firm  attachment  between  the  pile  shoe  and  the  interior  structure.    The  1200 


*  See  Deutsche  Bauzeitung,  1906,  Zementbeilage,  No.  21. 


PILES 


251 


kg.  (2645  lb.)  hammer  fell  with  a  1.3  to  1.5  m.  (4.2  to  5  ft.)  droj)  upon  an  oak 
block. 

In  Figs.  265-267  arc  shown  the  fabrication  of  the  reinforcement,  the  phicing 
of  the  concrete  in  the  horizontal  forms,  and  a  stock  of  complete  j^iles  with  some 
finished  units  of  reinforcement.  The  latter  were  built  by  placing  upon  a  l)ench 
the  spiral,  which  had  been  formed  upon  a  reel,  the  longitudinal  rods  were  then 
put  in  place  and  wired  to  every  second  or  third  turn  of  the  spiral.  The  concrete 
was  mixed  in  proportions  of  1:4^.    The  forms  were  so  arranged  that  the  side 


Fig.  265. — Factory  in  Metz.    Fabrication  of  pile  reinforcement. 

boards  could  be  removed  in  one  or  two  days,  while  the  pile  remained  ui)on  the 
bottom  piece  for  about  8  days. 

For  the  foundations  of  the  station  in  Cannstatt,  ])iles  from  5.5  to  10  m.  (18 
to  33  ft.)  long  were  driven  after  hardening  from  35  to  45  days.  Their  section  and 
reinforcement  were  exactly  like  those  of  Fig.  264.  The  driving  was  done  with  a 
Mench  and  Hambrock  steam  pile  driver.  'J'he  2750-kg.  (6050  lb.)  driving 
mechanism  consisted  of  a  ram  with  lifting  ro])e.  As  a  guide  for  the  wooden 
block,  a  wrought-iron  cap  was  used,  which  also  j)revented  a  splitting  of  the 
concrete,  because  it  completely  inclosed  the  pile  head.  Between  the  pile  and  the 
block  was  a  layer  of  sawdust  to  lessen  the  impact.  Concrete  piles  provided  with 
spiral  reinforcement  possess  such  impact  resistance  that  the  wooden  block  may 
be  entirely  omitted  and  the  hammer  allowed  to  strike  directly  on  the  concrete. 
It  may  crumble  the  concrete  of  the  head  somewhat,  as  is  shown  in  Fig.  270, 
for  which  no  cushion  was  used.    This  damaging  of  the  pile  top  is  without  danger, 


Fig.  267. — Behrend  Building,  Kiel.    Stock  of  finished  piles  and  some  reinforcement  units. 


PILES 


253 


Fig.  269. — Business  building  in  Metz.    Driving  piles. 


PILES 


255 


since  this  concrete  must  almost  always  l)e  removed  to  build  the  reinforced 
concrete  beam  or  column,  so  that  the  reinforcement  for  the  new  member  can 
be  joined  to  that  of  the  pile,  and  then 
all  concreted  together.  In  a  similar  way, 
reinforced  concrete  j^iles  can  be  spliced. 
In  the  foundations  of  the  station  in 
Cannstatt,  the  piles  were  driven  with 
a  hammer  fall  of  about  i  m.  (3.3  ft. 
approx.),  the  penetration   being  4-5  mm. 

to  ii^  )-  The  maximum  perform- 
ance of  the  driver  was  100  running  m. 
per  day   (328  ft.). 

In   computing    the    carrying  capacity 
of  the  piles,  the  Brix  formula  is  usually  ^'^^^  27o.-Station  in  Cannstatt. 

,       ,  Head  of  pile  which  was  driven  without 


wherein  h  is  the  fall  of  the  hammer; 

Q  the  weight  of  the  hammer; 
g  the  weight  of  the  pile; 

e  the  penetration  of  the  pile  under  the  last  blow 
p  is  double  the  safe  allowable  load  for  the  pile. 


The  quantity  e  will  naturally  be  the  average  of  the  last  few  blows. 
Fig.  269  shows  a  steam  pile  driver  at  work  on  a  new  business  building  in 
Metz. 


CHAPTER  XIII 


APPLICATIONS  OF  REINFORCED  CONCRETE 

BRIDGES 

(a)  With  Horizontal  Members.  Slab  Culverts. — In  the  early  stages  of 
railroad  construction,  culverts  roofed  with  natural  stone  were  extensively  em- 
ployed. With  the  advent  of  concrete  and  of  cement  pipe,  arched  conduits  easily 
constructed  in  concrete  or,  for  smaller  openings,  cement  pipes  were  substituted. 
With  the  introduction  of  reinforced  concrete,  however,  slab  culverts  again  became 


Fig.  271. — Laufbach  Bridge  in  Laupheim.    Placing  Reinforcement. 


useful.  Since  it  is  possible  with  the  aid  of  reinforcement  to  make  the  concrete 
slab  resist  any  bending  stress,  the  span  of  the  slab  or  the  clear  way  of  these 
culverts  can  be  increased  to  about  6.5  (21  ft.),  so  that  their  field  of  usefulness 
has  been  greatly  extended.  The  span  might  be  still  further  increased,  but 
beyond  about  5m.  (16  ft.),  T-beams  are  cheaper  than  simple  slabs. 

Slab  culverts  with  reinforced  concrete  covers,  are  used  over  railroads  as  well 
as  for  streets.  For  instance,  Wayss  &  Freytag  constructed  for  the  Gaildorf- 
Untergroningen  Railroad  a  whole  series  of  such  culverts,  and  in  the  station  at 

256 


BRIDGES 


257 


Soflingen,  a  foot  tunnel  of  this  same  sort.  •  Figs.  271-273  show  the  construction 
of  a  slab  culvert  of  4  m.  (13  ft.)  clear  span  under  the  market  ])lace  of  Laui)heim. 
The  35  cm.  (13.7  in.)  slab  was  calculated  for  a  steam  roller.  It  was  l)uilt  with 
12  rods,  16  mm.  (f  in.)  in  diameter  j)er  m.  width  (39.37  in.),  g  being  straight  and 
3  bent  upwards  at  the  ends.  Its  surface  is  crowned  to  discharge  surface  water. 
The  test  loading  showed  no  measurable  deilection.  Numerous  similar  slabs  under 
streets,  up  to  spans  of  6.5  m.  (22  ft.),  have  been  built. 


Fig.  272. — Laufbach  Bridge  in  Laupheim.  Cross-section. 


The  abutments  of  such  culverts  are  usually  built  of  concrete,  but  use  is  often 
made  of  existing  ones  of  masonry.  The  slab  covers  the  abutment,  thereby 
effecting  a  saving  in  wide  spans  through  a  saving  of  masonry.  The  slabs  are 
usually  constructed  at  the  site,  on  forms,  but  have  also  been  made  in  sections 
80  to  ICQ  cm.  (30  to  40  in.  approx.)  wide,  and  placed  after  hardening.  This 
is  necessary  where  no  interruption  to  traffic  can  be  allowed,  one-half  the  street 
being  first  constructed,  the  traffic  then  diverted  over  that  portion  while  the  other 
half  is  completed. 


I  I 

(   %  oo   ><-  o  so  -^^ 


Fig.  273. — Laufbach  Bridge  in  Laupheim.    Longitudinal  section  showing  thoroughfare. 


Under  railroad  embankments,  the  strength  of  the  reinforced  concrete  slab  can 
•always  be  suited  to  the  load,  by  reducing  the  thickness  towards  the  ends  of  the 
culverts.  Walls  over  the  ends  of  the  culvert  to  retain  the  fill  and  shorten  the 
length  of  the  masonry  work  can  advantageously  be  employed  and  anchored  to 
the  slab  by  the  cross  rods. 

Cantilevers. — Reinforced  concrete  slabs  can  be  employed  not  only  as  mem- 
bers between  two  supports  but  also  when  secured  at  one  end,  the  other  projecting 
freely.  This  form  of  construction  can  be  employed  for  instance  for  widening 
a  street  along  a  river.    The  sidewalk  can  then  be  allowed  to  project  over  the 


258 


CONCRETE-STEEL  CONSTRUCTION 


river,  as  in  Fig,  274.  Naturally  the  reinforcement  must  then  be  placed  near  the 
upper  surface,  and  be  anchored  in  a  block  of  concrete  behind  the  wall  of  the  em- 
bankment, the  block  being  of  sufficient  size  to  prevent  the  overturning  of  the 
walk.  Such  an  arrangement  was  built  by  Wayss  &  Freytag  at  Wildbad.  In  a 
similar  manner  the  footwalks  of  old  bridges  may  be  arranged  in  order  to  widen 
the  roadway. 

Reinforced  concrete  slabs  can  also  be  used  to  advantage  as  the  flooring  of 
steel  footbridges  and  viaducts,  and  also  for  the  construction  of  the  flooring  of 
the  main  thoroughfares  of  bridges  in  place  of  Zores*  iron  and  buckleplates. 
The  cost  of  such  structures  will  be  lessened  by  the  employment  of  reinforced 
concrete  and  in  all  cases  the  construction  will  be  simplified.  The  slabs  for 
sidewalks  are,  in  most  cases,  made  in  advance,  and  then  laid.    They  may  be 


<    —  zoo    ->K.„.^..-.  70  ...^  ..»» 


Fig.  274. — Cantilevered  sidewalk,  Schramberg. 


made  with  a  cement  surface  coat,  or  an  asphalt  wearing  surface  can  be  applied 
after  laying.  The  reinforced  concrete  construction  of  main  bridge  thorough- 
fares consists  of  continuous  reinforced  slabs  that  are  supported  either  between 
the  longitudinal  beams  or  usually  between  the  cross  beams,  and  are  reinforced 
according  to  the  maximum  moment  diagrams. 

T-Beam  Bridges. — For  larger  spans,  the  rectangular  slab  section  is  uneconom- 
ical, and  T-beams  are  usually  employed  for  the  carrying  members  of  spans  exceed- 
ing about  5  m.  (16  ft.).  The  usual  arrangement  is  to  span  the  opening  with 
several  similar  parallel  girders  and  lay  a  floor  slab  between  them.  With  ref- 
erence to  the  concentrated  load  of  a  steam  roller,  a  girder  spacing  of  between  1.3 
and  1.6  m.  (4.2  to  5.2  ft.)  should  be  used,  and  for  the  same  reason,  and  because 
of  the  unequal  deflection  of  the  girders,  the  floor  slabs  should  have  straight  rods 
the  full  width  both  above  and  below,  as  well  as  bent  ones,  and  numerous  dis- 
tributing rods. 

Such  a  T-beam  bridge  vv^ith  a  clear  span  of  12.07  (39-6  ft.)  (the  Horn- 
bach  Bridge  at  Zweibriicken)  is  shown  in  Figs.  275-276.    The  bridge  is  skew^ 

*  Z-bars,  Phoenix  column  shapes,  and  some  other  similar  sections. — (Trans.) 


BRIDGES 


259 


and  rests  partly  on  pre-existing  stone  masonry  abutments.  The  girders  are 
connected  by  cross-beams,  which  serve  to  distribute  more  uniformly  over  several 
girders  the  concentrated  loads  and  those  of  the  brackets  on  which  the  reinforced 


Fig.  275. — Hornback  Bridge  near  Zweibriicken. 

concrete  slabs  of  the  sidewalk  are  supported.  As  reinforcement,  the  girders 
have  five  straight  round  rods  of  30  mm.  (13%  in.),  and  five  bent  ones  of  28  mm. 
(ij  in.  approx.)  diameter,  the  bending  of  the  latter  being  arranged  to  care  for 


Fig.  276. — Hornback  Bridge  near  Zweibrucken.    Cross  and  longitudinal  sections. 

the  diagonal  tensile  stresses  produced  by  the  shearing  forces.  Under  the  test 
load  of  a  20  t.  (22  ton)  steam  roller,  the  girders  were  deflected  only  0.3-0.4 
mm.  (0.015 


260 


CONCRETE-STEEL  CONSTRUCTION 


T-beam  bridges  of  this  type,  of  spans  up  to  i6  m.  (52  ft.)  are  entirely  prac- 
ticable, and  in  m^st  cases  cheaper  than  steel  bridges.  Special  instances  occur 
of  20  m.  (66  ft.)  spans,  and  unconnected  girders  also  exist,  such  as  are  shown 
in  Fig.  277.  With  longer  spans,  the  girders  become  rather  heavy,  so  that  T- 
beam  bridges  possess  little  superiority  over  steel  ones. 

In  narrow  bridges,  up  to  6  ra.  (20  ft.)  width,  less  depth  is  involved  when  only 
two  parallel  girders  are  employed,  and  the  weight  of  the  roadway  is  transferred 
to  them  by  cross  beams.  Such  small  depth  is  important  where  railroads  have 
to  replace  grade  crossings  by  bridges.  An  example  of  such  a  case  is  shown  in 
Figs.  278-281,  of  a  bridge  at  Grimmelfingen,  near  Ulm.  The  girders  with  a 
clear  span  of  9  m.  (30  ft.)  have  a  rectangular  section  70  cm.  (27.6  in.)  wide,  and 


Fig.  277. — Bridge  near  Krapina,  with  open  girders  of  200  m.  (66  ft.)  span. 

extend  up  above  the  roadway,  thus  forming  a  low  parapet,  upon  which  only  a 
small  railing  is  necessary.  The  reinforced  concrete  slabs  which  carry  the  road- 
way and  span  the  1.533  (S  ^^•)  spaces  between  beams,  are  covered  with  a  water- 
tight coating  of  layers  of  asphalt  felt.  Ready  drainage  is  effected  not  only  by  a 
slope  toward  the  girders  from  the  center  of  the  roadway,  but  also  by  a  slope 
toward  the  abutments  from  the  center  of  span.  The  roadway  proper  is  con- 
structed of  concrete. 

The  static  computation  was  made  for  a  load  consisting  of  a  crowd  of  people 
of  450  kg/cm-  (92  lbs/ft^)  and  a  6  t.  (6.6  ton)  wagon  with  a  1.5  t.  (1.65  ton)  wheel 
load.  The  first  loading  determined  the  girders,  while  the  latter  affected  the  deck 
slabs  and  the  floor  beams.  The  calculation  and  dimensioning  of  the  deck  was 
based  on  the  assumptions  of  continuity  throughout  the  several  panels,  and  of  free 
support  of  the  beams.  The  reinforcement  consisted  of  seven  round  rods  of  7 
mm.  (yq  in.  approx.)  diameter  per  meter  width,  above  and  below,  throughout, 


262 


CONCRETE-STEEL  CONSTRUCTION 


and  of  seven  bent  rods,  lo  mm.  (f  in.  approx.)  diameter,  bent  upward  in  the 
vicinity  of  the  beams  and  carried  over  them. 

The  cross  beams  were  calculated  without  regard  to  possible  end  restraint. 
At  the  same  time,  however,  the  arrangement  of  the  reinforcement  provides  some 
rigidity  at  their  connections  with  the  girders.  Of  the  four  rods  (26  mm. — 13^  in. 
approx.)  required  at  the  centers  of  the  beams,  two  are  bent  upward  near  the 
girders,  at  points  where  their  area  is  not  required  in  the  lower  chord. 

The  two  girders  are  not  anchored  at  the  abutments  and  consequently  must 
be  considered  as  freely  supported  beams  of  rectangular  section.    Of  the  ten 


Fig.  282. — Highway  Bridge  in  Grimmelfingen  near  Ulm.    Tested  to  450  kg/m^  (92  lbs/ft^) 
with  gravel;  deflection  0.2  mm.  (o.ooS  in.). 


round  rods,  each  of  30  mm.  (if  in.  approx.)  diameter  required  at  the  center,  at 
special  points  six  are  bent  upward  at  an  angle  of  45°  to  resist  in  the  most  effec- 
tive manner  the  shearing  or  diagonal  tensile  stresses.  Naturally,  the  bending 
is  done  at  points  where  the  moment  is  sufficiently  reduced.  The  bent  ends  are 
carried  over  the  supports  so  that  possible  reverse  moments  may  be  resisted. 

The  zone  of  compression  of  the  girders  is  reinforced  with  three  round  rods 
18  mm.  {{^  in.)  in  diameter,  connected  together  and  anchored  in  the  concrete 
of  the  girders  by  7  mm.  (J  in.  approx.)  stirrups.  They  strengthen  the  upper 
side  of  the  girder  against  compressive  stresses. 

An  important  advantage  of  such  reinforced  concrete  bridges  over  railways, 
is  that  they  are  not  affected  by  the  gases  from  the  locomotives,  which,  in  the  case 
of  busy  stretches  of  track,  and  where  difficult  of  access,  cause  active  corrosion 
and  high  maintenance  charges  for  steel  structures.    Fig.  283  gives  an  instruc- 


BRIDGES 


263 


tive  illustration  of  the  destruction  wrought  by  smoke  gases  in  unfavorable  condi- 
tions.   The  piece  there  shown  was  removed  early  in  1907  from  a  steel  longitu- 
dinal girder  under  the  roadway  of  a  bridge  over  a  freight  station  erected  in  1886. 
If  the  length  of  a  horizontal  reinforced  concrete  bridge  is  greater  than  16  to 


I 

Fig.  283. — Rusting  of  a  main  girder  by  locomotive  gases,  ^ 

20  m.  (43  to  66  ft.),  intermediate  supports  must  be  provided.  They  may  consist 
of  ordinary  masonry  intermediate  piers,  especially  where  such  may  have  been 
left  standing  from  a  previous  wooden  bridge;  but  usually,  however,  are  made 
of  reinforced  concrete  in  the  shape  of  columns.  It  is  best  to  place  a  separate 
support  under  each  girder,  and  to  connect  the  columns  at  the  base  by  means  of 
a  common  pedestal  and  a  single  foundation. 


Fig.  284. — Reinforced  concrete  bridge  over  a  railroad  cut. 


Continuous  members  with  three  spans  are  well  adapted  for  bridging  railroad 
cuts  (Fig.  284),  a  considerable  saving  being  effected  in  abutment  masonry. 

In  Fig.  285  are  shown  the  entire  design  and  details  of  a  section  through  the 
thoroughfare  of  such  a  bridge  over  a  ravine  at  Bad  Tolz.  In  this  case  the  abut- 
ments are  carried  down  through  the  shifting  soil  to  bedrock  on  tubular  piles  of 


264 


CONCRETE-STEEL  CONSTRUCTION 


reinforced  concrete.  The  reinforcement  of  the  continuous  bridge  girders  is 
designed  exactly  like  those  of  buildings.    Fig.  286  shows  a  view  of  this  bridge. 


Fig.  285. — Bridge  over  a  ravine  near  Bad  Tolz. 


In  Straight  girder  bridges  of  greater  length  an  expansion  joint  must  be 
provided  about  every  fourth  opening.     This  necessitates  cutting  one  of  the 


Fig.  286. — Bridge  over  a  ravine  near  Bad  Tolz. 


piers  longitudinally,  or  constructing  two  piers  close  together  and  allowing  the 
girders  to  project  over  them  as  cantilevers.  Bracketed  beam  ends  may  often  be 
employed  to  advantage  in  reinforced  concrete  bridge  construction.    Pains  should 


BRIDGES 


265 


be  taken  to  arrange  the  supports  of  reinforced  concrete  bridges  so  that  the  j)res- 
sures  at  those  points  are  properly  carried.  In  small  bridges,  sliding  and  tangen- 
tial tilting  supports  are  suitable,  roller  bearings  being  useful  only  in  exceptional 
cases.  Practice  has  indeed  not  yet  shown  the  absolute  necessity  of  such  devices 
for  reinforced  concrete  construction,  since  at  such  points  much  smaller  move- 
ments take  place,  and  where  girders  are  supported  by  columns  such  devices 
are  of  small  moment,  because  of  the  elasticity  of  the  column.  Where  necessary^ 
the  secondary  stresses  so  produced  can  be  calculated.  With  beams  of  only  a 
single  span,  dangerous  though  invisible  conditions  exist  with  rigid  supports, 
since  the  bottom  of  the  beam  lengthens  from  the  tensile  stresses  which  appear 


Fig.  287. — Cover  of  the  lU-Hochwasser  Canal,  Miilhausen. 


when  the  forms  are  removed,  and  the  beam  exerts  a  pressure  on  the  abutments, 
which  is  bad  for  both  beams  and  wall.  In  continuous  beams,  however,  which 
rest  on  masonry  center  piers,  such  an  arrangement,  when  j)roperly  constructed, 
is  advisable. 

All  varieties  of  deck  structures  are  identical  in  jjrinciple  with  T-beani  bridges, 
especially  those  over  streams  and  railroads.  In  those  over  streams  it  often 
happens  favorably  that  existing  river  walls  may  be  used  as  abutments  for  shear- 
free  constructions.  In  covering  railroad  cuts,  the  principal  advantage  of  rein- 
forced concrete  is  its  capacity  to  withstand  the  locomotive  gases. 

An  extensive  cover  of  this  kind  in  Miilhausen,  in  Alsace,  over  the  Ill-Hoch- 
wasser  canal  is  shown  in  Fig.  287.  The  work  illustrated  includes  a  deck  36  m. 
(118  ft.)  wide  and  660  m.  (2165  ft.)  long.  The  beams,  spaced  3  m.  (9.8  ft.)  apart, 
were  continuous  over  spans  of  11,  14  and  11  m.  (36,46  and  36  ft.)  with  ends 


ARCHES 


267 


elastically  restrained  l)y  reinforced  concrete  columns  in  the  side  walls.  They 
were  designed  with  that  end  in  view  and  were  computed  for  a  live  load  of  500 
kg/m-  (102  lbs  ft-).  Since  these  reinforced  concrete  columns,  which  had  also 
to  resist  the  earth  pressure,  were  inclosed  in  the  wall  construction,  it  was  possible 
to  construct  the  girders  with  less  depth.  The  7  m.  (23  ft.)  high,  octagonal, 
intermediate  supporting  columns  rested  on  the  reinforced  concrete  piles  described 
previously  (Fig.  264).  Between  the  columns,  5  m.  (16  ft.)  high  reinforced 
concrete  walls  were  erected,  so  that  at  high  water  it  llowcd  in  three  separate 
streams.  These  partitions  afforded  a  stronger  j)rotection  for  the  columns  and 
a  considerable  stiffening  of  the  construction  in  a  longitudinal  direction. 

Four  streets  crossed  the  canal,  and  it  was  because  of  the  high  loads  from 
street  cars  and  heavy  trucks  that  this  s])ecially  strong  construction  was  re(|uircd. 

The  largest  piece  of  work  of  this  description  is  that  of  the  Vienna  Municipal 
Railroad,  which  extends  2  km.  (1.24  miles)  with  spans  of  12.7  m.  (41.6  ft.),  and 
was  constructed  by  G.  A.  Wayss  &  Co.,  of  Vienna. 

(/;)  Arch  Construction. — In  arched  bridges,  reinforced  concrete  can  be  em- 
ployed either  for  the  arch  alone,  or  the  superstructure  including  the  roadway, 
or  in  all  structural  parts.  In  small  spans,  the  reinforcement  in  the  arch  enables 
the  full  compressive  strength  of  the  concrete  to  be  utilized,  since  the  tensile 
stresses  are  independently  resisted.  In  medium  spans  of  40  to  50  m.  (130  to 
165  ft.),  the  employment  of  reinforced  concrete  as  the  arch  material  is  less  frequent, 
since  in  this  case,  provided  a  proper  profile  has  been  employed,  no  tensile  stresses 
occur,  because  of  the  large  dead  load.  On  the  other  hand,  reinforced  concrete 
is  better  adapted  for  long  spans.  If  the  safe  compressive  stress  in  the  arch 
is  not  to  be  exceeded,  it  is  necessary  to  limit  the  weight  of  the  superstructure, 
and  this  can  be  done  by  a  suitable  employment  of  reinforced  concrete.  In 
this  way  the  dead  load  stresses  will  be  greatly  reduced,  but  the  small  edge  stress 
may  decrease  to  zero,  or  change  to  tension  under  unfavorable  live  loads,  so 
that  reinforcement  is  again  necessary. 

In  consequence,  where  it  is  desired  in  small  and  medium  spans  that  no  tensile 
stresses  shall  exist,  or  in  other  words  where  only  concrete  work  is  employed, 
the  superstructure  cannot  be  kept  too  light.  A  light  superstructure  is  then 
justifiable  only  when  demanded  for  architectural  reasons  or  when  it  is  necessary 
to  impose  as  little  weight  as  possible  on  the  foundations. 

An  example  of  a  reinforced  concrete  bridge  without  special  superstructure 
is  shown  in  Fig.  288.  The  arch  of  36  m.  (118  ft.)  span  and  4.2  m.  (13.8  ft.)  rise, 
is  50  cm.  (20  in.)  thick  at  the  crown  and  was  computed  as  fixed  at  the  ends.* 
The  reinforcement  consisted  of  ten  14  mm.  in.)  rods  near  both  the  top  and 
the  soffit  of  the  arch,  and  at  distances  of  about  50  cm.  (20  in.),  the  two  systems 
of  rods  were  tied  together  by  7  mm.  in.  a])prox.)  stirrups.  At  the  spring- 
ings  the  arch  rested  with  a  widened  foot  upon  the  abutment  concrete  so  as  to 
secure  the  assumed,  computed  restraint.  The  arch  is  faced  with  ashlar  masonry. 
The  space  between  the  asphalt  waterproofed  top  of  the  arch  and  the  roadway 
is  filled  with  gravel,  upon  which  rests  the  concrete  foundation  of  the  asphalt 
street  surface.    The  arrangement  of  the  arch  centers  and  other  details  is  also 

*  See  article  by  the  author  published  in  the  "  Schweizerischer  Bauzeitung,"  1908,  Nos.  7 
and  8;  and  also  the  separate  brochure,  "  Berechnung  von  eingespannter  Gewolben." 


268 


CONCRETE-STEEL  CONSTRUCTION 


shown  in  the  illustration.  The  maximum  stresses  amounted  to  37.4  kg/cm^ 
(532  lbs/in^)  compression,  and  o.i  kg/cm^  (1.4  lbs/in^)  tension,  the  reinforce- 
ment being  thus  only  a  safeguard  in  case  an  abutment  should  settle  and  cause 
an  increase  in  the  tensile  stresses. 

During  the  last  twenty  years  a  large  number  of  bridges  similar  to  this  have 
been  erected  by  Wayss  &  Freytag. 

With  regard  to  the  application  of  reinforced  concrete  to  the  construction  of 
the  roadway  and  the  superstructure  over  the  arch,  several  arrangements  are 
possible. 

I.  The  reinforced  concrete  slab  which  carries  the  roadway  may  rest  on  40 
to  60  cm.  (16  to  24  in.)  walls  of  concrete  or  masonry,  which  are  combined  with 
the  arch  and  carry  the  loads  to  it.  The  arches  can  be  constructed  either  re- 
strained or  three  hinged  by  using  suitable  material,  entirely  independent  of  the  rein- 


FiG.  289. — Highway  bridge  near  Hamburg.    Longitudinal  and  horizontal  sections. 


forced  concrete  construction  of  the  roadway.  The  reinforced  slabs  of  the  latter 
correspond  with  the  spandrel  arches  of  the  usual  arrangement,  but  are  more 
advantageous,  because  they  may  extend  somewhat  over  the  outside  walls  and 
because  no  horizontal  shear  is  produced  by  spandrel  walls.  Consequently,  a 
narrower  arch  is  possible,  with  a  saving  in  the  abutments  and  piers.  When 
spandrel  walls  of  considerable  height  are  used,  it  is  wise  to  stiffen  them  with 
intermediate  decks.  A  bridge  of  this  arrangement  of  superstructure  is  illus- 
strated  in  Fig.  289.  The  arch  of  that  bridge  consists  of  mass  concrete  with- 
out reinforcement.  Since  the  ground  behind  the  abutments  slopes  up  to  the  level 
of  the  roadway,  the  longitudinal  walls  with  the  reinforced  concrete  slab  resting 
on  them  could  be  extended  into  the  ground  over  the  abutments,  so  that  wing 
walls  and  embankments  were  unnecessary.  In  this  manner  a  saving  in  cost 
could  be  accomplished. 


ARCHES 


269 


2.  Over  the  arch,  and  at  right  angles  to  its  face,  cross  walls  can  be  erected 
to  support  the  reinforced  concrete  construction  of  the  roadway.  These  cross 
walls  are  usually  j)laced  at  such  a  distance  apart  that  a  continuous  reinforced 
concrete  slab  may  l)e  provided  for  the  supi)ort  of  the  roadway.  The  outside 
walls  thus  fall  entirely  beyond  the  arch,  and  the  latter  does  not  receive  as  much 
stiffening  from  the  superstructure  as  in  the  last  case,  so  that  its  form  must  be 
accurately  designed  and  executed.  Its  superiority  from  a  static  point  of  view 
rests  in  the  decreased  load  on  the  abutments  and  foundations. 

The  arrangement  with  cross  walls  is  very  light  and  j)leasing  in  appearance. 
The  walls  may  be  constructed  of  masonry,  or  of  concrete,  with  or  without  rein- 
forcement. The  latter  kind  of  construction  permits  a  narrow  width  to  be  em- 
ployed and  may  be  used  when  the  arch  consists  of  reinforced  concrete. 


Fig.  290. — Highway  bridge  near  Hamburg.    Section  through  superstructure  and  thoroughfare. 

In  Fig.  291  is  illustrated  a  foot  bridge  over  the  canal  near  the  Grosshesseloher 
railroad  bridge.  The  pleasing  appearance  is  secured  without  employing  architec- 
tural adornment,  but  solely  through  the  structural  work,  all  conspicuous  parts 
of  which  consist  of  reinforced  concrete. 

A  highway  bridge  of  the  same  type  is  shown  in  ¥ig.  292.  The  bridge  has  a 
32  m.  (105  ft.)  span,  and  was  erected  on  the  site  of  a  dilapidated  wooden  one. 

In  Figs.  293-294  are  illustrated  a  foot  bridge  on  the  Metz-Vigy  line.  The 
axes  of  the  ribs  in  these  bridges  were  assumed  as  parabolas,  so  as  to  be  able  to 
apply  advantageously  available  formulas  for  the  immediate  computation  of  the 
influence  line  for  the  load  point  moments  of  a  restrained  parabolic  arch.  Since 
usually  the  axes  of  restrained  arches  are  assumed  so  as  to  coincide  with  the  line 
of  pressure  for  dead  load,  in  the  foregoing  cases  the  assumption  of  a  parabola  was 
permissible,  because  the  dead  load  is  small  and  very  nearly  uniformly  distributed 
over  the  span.  It  is  to  be  especially  noted  that  here  the  connection  of  the 
structure  with  the  bank  is  effected  by  means  of  concrete  walls  which  inclose  a 
hollow  space  decked  over  with  a  reinforced  concrete  slab.  In  this  way  the 
earth  pressure  is  diminished  against  the  wing  walls. 


Fig.  292. — Highway  bridge  in  Gunzesried  (Allgan).    Span  32  m.  (105  ft.). 


ARCHES 


271 


3.  The  cross  walls  in  the  foregoing  arrangement  may  be  replaced  with  trans- 
verse rows  of  columns  which  carry  the  T-beam  construction  of  the  roadway. 
In  this  way  the  weight  of  the  superstructure  over  the  arch  will  l^e  reduced  as 


Fig.  293. — Foot  bridge  over  the  Metz-Vigy  Line.    Test  load. 


much  as  possible,  this  arrangement  being  adapted  for  large  spans.  The  best 
example  to  date  of  this  class,  is  the  Isar  bridge  in  Grunwald  (Figs.  295-304),  a 
short  description  of  which  will  here  be  given.* 


Fig.  294. — Foot  bridge  over  the  Metz-Vigy  Line.    Longitudinal  section. 


This  bridge,  which  spans  the  Isar  between  Hollriegelsgreuth  and  Grunwald, 
was  built  by  the  reinforced  concrete  companies*  of  Munich  after  the  designs  of 

*  See  article  by  the  author  in  the  "  Schweizerische  Bauzeitung,"  1904,  XLIV,  Nos,  23  and 
24,  also  published  separately.    Figures  296  to  304  are  borrowed  from  that  publication, 
t  Wayss  &  Freytag  of  Neustadt-on-the-Hardt,  and  Kallmann  &  Littmann  of  Munich. 


•272 


CONCRETE-STEEL  CONSTRUCTION 


ARCHES 


273 


the  author.  This  highway  bridge  is  of  about  220  m.  (722  ft.)  length,  with  two 
arches  of  70  m  (229.6  ft.)  span,  each  with  a  rise  of  12.8  m.  (42.0  ft.)  over 
the  beds  of  the  river  Isar  and  the  power  canal  for  the  electric  plant.  These 
two  main  spans  are  finished  on  the  right  by  a  single,  and  on  the  left  by  four 
approach  s})ans  8.5  m.  (27.9  ft.)  long,  which  are  decked  with  a  straight  reinforced 
concrete  structure. 

The  two  main  spans  were  designed  as  three-hinged  arches.  In  determin- 
ing the  selection  of  this  form  of  construction,  besides  its  general  excellence,  one 
main  condition  also  existed,  that  up  to  the  date  of  the  arrangement  of  this  pro- 
ject, no  positive  data  existed  concerning  the  underlying  conditions,  so  that  much 
caution  was  necessary. 


Fig.  295. — Highway  bridge  over  the  Isar  near  Griinwald. 


The  bridge  was  designed  for  a  crowd  of  people  of  400  kg/m^  (82  lbs/ft^), 
and  a  steam  roller  load  of  20  t.  (22  tons).  As  described  by  the  author  in  the 
Zeitschrift  fur  Architektur  and  Ingenieurwesen,"  No.  II,  1900,  the  arches 
were  so  proportioned  as  to  form  and  thickness  that  at  each  section  the  maximum 
stresses  in  the  upper  and  lower  edges  were  equal  to  the  one  then  considered 
permissible  of  35  kg/cm^  (497  lbs/in^). 

At  the  crown,  the  arch  thickness  was  75  cm.  (29.5  in.),  at  the  springing  90 
cm.  (35.4  in.),  and  at  joints  V  and  VI*  it  reached  1.20  m.  (47.2  in.),  its  greatest 
thickness.  No  tensile  stresses  appeared,  but  the  comj^ression  was  reduced 
to  2.1  kg/cm^  (29.8  lbs/in^),  with  unfavorable  loading. 

For  each  centimeter  (0.4  in.)  which  the  arch  rib  deflected,  the  edge  stress 
varied  about  i  kg/cm^  (14.2  lbs/in^),  so  that  with  a  deflection  of  4  to  5  cm. 


*  Under  the  verticals  counted  from  the  abutments. —  (Trans.) 


274 


CONCRETE-STEEL  CONSTRUCTION 


(1.5  to  2  in.  approx.),  tensile  stresses  would  exist  in  sections  IV- VI.  Since 
such  a  deflection  did  not  appear  impossible  through  uncertainty  of  construc- 
tion, or  unequal  lowering  of  the  centers,  the  arch  was  provided  with  a  rein- 
forcement, which  obviously  was  impossible  of  computation,  and  was  introduced 
only  as  an  additional  measure  of  security.  This  reinforcement  consisted  of 
nine  round  rods  28  mm.  (ij  in.  approx.)  in  diameter,  both  above  and  below 
over  the  whole  width  of  8  m.  (26.2  ft.).  At  distances  of  i  m.  (40  in.)  the  upper 
and  lower  steel  was  tied  together  by  7  mm.       in.  approx.)  round  iron  stirrups. 

The  hinges  of  cast  steel  w^re  of  the  dimensions  shown  in  Figs.  297-298, 
and  their  smoothed  faces  rested  agahist  squared  reinforced  artificial  stones  with 
a  4  mm.  in.)  layer  of  sheet  lead  between.  The  pressure  over  the  bearing 
area  was  100  kg/cm^  (1422  lbs/in^).    The  hinges  were  designed  for  a  permissible 


Fig.  297. — The  abutment  hinge.  Fig.  298. — The  crown  hinge. 


bending  stress  of  11 00  kg/cm^  (15,642  lbs/in-),  and  the  two  halves  met  on 
cylindrical  surfaces  of  250  and  200  mm.  (9.84  and  7.87  in.)  radii.  Granite  was 
first  proposed  for  the  hinge  bearing  blocks,  but  later,  reinforced  concrete  blocks 
were  selected  because  more  economical,  and  after  compression  experiments 
made  in  Miinich  had  disclosed  good  results.  The  hinge-bearing  blocks  are 
under  pressure  over  only  a  part  of  their  area.  Similarly  loaded  test  specimens 
failed  by  cracking  in  the  direction  of  stress,  so  that  a  reinforcement  was  neces- 
sary, running  at  right  angles  to  the  direction  of  stress  and  transverse  to  the 
pressure  areas,  in  order  to  increase  sufficiently  the  ultimate  strength. 

In  constructing  the  blocks,  which  were  molded  in  perfect  cast-iron  forms  with 
smooth  surfaces,  the  reinforcement  was  uniformly  distributed  throughout  the 
full  height,  since  the  cracks  had  appeared  in  the  test  specimen  in  the  part  between 
the  pieces  of  reinforcement.    The  length  of  each  block  was  79  cm.  (31  in.)  and 


ARCHES 


275 


corresponded  with  the  length  of  a  hinge  piece,  so  that  a  single  piece  rested  on  each 
block.  At  the  abutment  hinges  the  i)arts  were  so  arranged  that  both  pieces  of 
stone  rested  on  the  centers,  thus  making  impossible  anything  except  a  simultaneous 
shifting. 

The  arch  was  concreted  in  separate  sections,  the  size  and  order  of  which  is 
shown  in  Fig.  296.    In  making  this  arrangement,  care  was  taken  concerning 


Fig.  299. — Detail  of  the  forms  of  the  Isar  Bridge. 


the  centering  that  the  largest  sections,  1-6,  usually  covered  a  whole  panel,  with 
a  small  space  left  free  at  the  ends  of  the  panels  over  the  posts,  and  also  a  small 
space  next  the  hinge  stones.  For  the  order  of  concreting  the  small  sections,  7-14, 
it  was  specified  that  those  next  the  hinges  should  be  done  last,  so  that  only  hght 
loads  could  come  upon  them,  and  thus  prevent  the  development  in  them  of  a 
dangerous  deformation.  The  last  sections  were  those  immediately  adjacent  to  the 
hinge  stones  in  the  abutments.  The  arch  was  built  with  a  mixture  of  i  part 
Blaubeur  Portland  cement,  2  parts  Isar  sand,  and  4  parts  Isar  gravel. 


276 


CONCRETE-STEEL  CONSTRUCTION 


The  forms  for  both  main  spans  consisted  of  seven  sections  of  the  construc- 
tion shown  in  Fig.  296.  By  this  arrangement,  it  was  insured  that  the  support- 
ing of  the  vertical  loads  from  the  concrete  construction  would  be  done  most 
directly  by  the  piles,  so  that  the  panels  would  be  the  only  construction  parts- 
subjected  to  bending  stresses.  In  this  way  deformation  of  the  scaffolding  was 
reduced  to  a  minimum,  and  to  this  end,  as  far  as  precautionary  measures  could 
be  taken,  the  posts  and  braces  were  not  allowed  to  bear  directly  against  the 
wood  of  the  sills,  that  is,  so  that  the  latter  would  be  overstressed  in  a  direction 
at  right  angles  to  the  fibers;  13  to  15  kg/cm^  (185  to  213  lbs/in^)  was  assumed 
as  a  safe  permissible  stress  on  timber  in  that  direction,  and  pieces  of  channel  iron 
were  employed  to  distribute  the  pressures  of  the  posts  and  piles  over  the  sills 
and  cross-beams  of  the  scaffolding.    (Fig.  299.) 


Fig.  300. — Highway  bridge  over  the  Isar  near  Griinwald.    View  of  the  left  abutment. 


Furthermore,  sand  boxes  were  employed,  except  for  the  first  range  from  the 
abutment,  where  wedges  were  used.  Compared  with  screw  jacks,  a  saving  in 
cost  was  effected;  and  further,  the  sand  boxes  offered  the  added  advantage  of 
a  stabler  support  for  the  scaffolding,  and,  with  sufficient  caution  and  experience, 
as  safe  a  form  of  centering  was  secured  as  with  the  average  screw  jack. 

All  foundations  of  abutments  and  piers  were  carried  down  by  pumping  to 
the  rock  (a  kind  of  marl),  which  could  be  loaded  to  5  kg/cm^  (5  tons/ft^).  The 
highest  pier  of  the  approach  spans  and  the  superstructure  of  the  principal  pier 
contained  open  spaces,  limited  in  width  by  the  reinforcement  of  the  upper 
thoroughfare  arches  on  these  piers,  and  the  condition  that  the  floor  beams  were 
given  sufficient  bearing  surface. 

The  bridge  floor  had  a  breadth  of  8  m.  (26.2  ft.)  between  the  side  rails,  of 


ARCHES 


277 


which  5  m.  (16.4  ft.)  was  given  to  the  roadway  and  1.5  m.  (4.9  ft.)  on  each  side 
to  a  sidewalk.  The  floor  sloped  from  the  center  pier  in  each  direction  on  a  1% 
grade  and  was  drained  through  the  piers,  over  the  abutments  of  each  main  span. 
The  roadway  was  carried  by  a  reinforced  concrete  construction,  consisting  of  a 
reinforced  slab  8.6  m.  (28.2  ft.)  wide  and  20  cm.  (7.9  in.)  thick,  which  trans- 
ferred its  load  to  five  longitudinal  beams  25X40  cm.  (9.8X15.7  in.)  in  section^ 
which  in  turn  were  supported  from  the  arch  by  reinforced  concrete  columns  4  m. 
(13. 1  ft.)  apart.  The  slab  and  the  beams  were  designed  as  continuous  mem- 
bers, in  which  the  unfavorable  assumption  was  made  that  the  slab  was  free  to 
move  on  the  beams  and  the  latter  on  the  columns.  For  the  calculation  of  the 
roadway  construction  the  wheel  load  of  a  steam  roller  was  critical.  The 
reinforcements  of  the  slabs  and  beams  are  shown  in  Figs.  303  and  304.    Over  the 


Fig.  301. — View  under  the  arch  over  the  right-hand  stream. 


columns,  the  section  of  each  beam  was  increased  by  haunches,  so  that  the  com- 
pressive stress  on  the  underside,  because  of  a  large  negative  end  moment,  did 
not  exceed  safe  limits.    These  haunches  also  reduced  the  shearing  stresses. 

The  columns  have  a  section  of  40X40  cm.  (15.7  in.)  with  the  exception  of 
those  in  plain  sight  at  the  sides  of  the  bridge,  which  were  given  a  T-section  to 
improve  their  appearance,  so  that  they  had  a  breadth  of  70  cm.  (27.6  in.)  on 
the  outside.  The  reinforcement  of  the  longest  columns  consisted  of  eight  round 
rods  24  mm.  (Jf  in.)  in  diameter,  while  the  succeeding  rows  had  eight  rods  of 
22  mm.  (I  in.),  four  rods  of  24  mm.  (jf  in.),  and  four  rods  of  22  mm.  (Jin.) 
diameter  respectively;  and  the  outside  columns  were  reinforced  with  from  eight 
rods  20  mm.  (xf  in.)  to  four  rods  18  mm.  (xJ  in.  approx.)  in  diameter.  In  all 
columns,  a  tie  spacing  of  35  cm.  (13.8  in.)  was  used,  for  the  7  mm.  (^  in.  approx.) 
round  wires  employed.    The  column  steel  extended  about  40  to  50  cm.  (16  to  20 


278 


CONCRETE-STEEL  CONSTRUCTION 


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in,)  into  the  arch  concrete,  and  under  each  row  of  columns,  the  arch  was  rein- 
forced laterally  by  four  i6  mm.  (f  in.)  round  rods  below  and  two  similar  rods 
above,  so  as  better  to  distribute  the  concentrated  loads  of  the  columns  over  the 
whole  arch  width.  The  last  sup})ort  over  the  abutments  was  built  as  a  rein- 
forced concrete  wall,  with  openings,  so  as  to  provide  the  necessary  lateral  stability 
for  the  thoroughfare  deck.  Over  the  hinges  at  crown  and  abutments,  expansion 
joints  were  provided  in  the  deck  construction,  the  joints  being  covered  with 
sheet  metal  in  the  usual  way. 

The  reinforced  concrete  construction  over  the  8.5  m.  (27.9  ft)  wide  approach 
spans  consisted  of  a  deck  slab  and  girders.  Since  the  girders  had  the  same 
spacing  as  those  over  the  main  spans,  the  slab  was  built  exactly  like  that  one. 
The  beams  were  designed  and  constructed  like  simple,  freely  supported  mem- 


FiG.  303. — Reinforcement  of  the  deck  slab. 


bers,  so  as  to  simplify  the  reinforcement  which  consisted  of  five  round  rods  36 
mm.  (i^  in.)  and  one  rod  24  mm.  (}f  in.)  in  diameter.    (See  Fig.  304). 

The  architecture  of  the  bridge  is  completely  determined  by  its  construction. 
With  the  exception  of  the  center  one,  no  pier  is  at  all  decorated.  All  concrete 
surfaces  were  left  without  manipulation,  except  one  prominent  ridge  which  was 
formed  by  a  crack  between  the  form  boards.  The  railing  was  also  constructed 
of  concrete  with  open  panels,  without  extra  finish.  The  water-tight  covering  of 
the  concrete  slab  which  carried  the  thoroughfare  consisted  of  a  layer  of  asphalt, 
that  of  the  arches  being  a  water-tight  cement  coating. 

When  the  centers  were  struck,  the  concrete  was  about  three  months  old, 
and  the  whole  dead  load,  including  the  pavement,  was  in  place,  so  that  the 
abutment  pressure  had  exactly  the  direction  computed  for  it. 

The  striking  of  the  centers  was  so  arranged  that  first,  at  a  given  signal,  the 
sand  boxes  were  opened  under  both  middle  sections  beneath  the  center  joint, 
and  1 1.  (J  pint)  of  sand  allowed  to  run  out.  The  opening  was  then  closed,  and 
a  couple  of  blows  struck  upon  the  sand  box  which  caused  a  settlement  of  a  few 
millimeters.  The  same  operation  was  repeated  simultaneously  on  the  next  four 
rows  of  supports  each  side  the  center,  and  so  on,  to  the  third  series,  after  which  the 
process  from  the  crown  outward  w^as  repeated,  and  all  except  the  last  row  was 


280 


CONCRETE-STEEL  CONSTRUCTION 


lowered.  A  total  of  twenty-eight  men  with  the  requisite  inspection  force  was 
necessary,  each  one  being  equipped  with  a  wrench,  an  ax,  a  measuring  vessel, 
and  a  mallet.  Since  the  centering  was  itself  strained  elastically,  only  a  very  small 
deflection  of  the  arch  was  observed.  Since  the  deflection  was  not  larger,  the 
oak  wedges  next  the  abutments  were  also  loosened. 

When  a  lowering  of  the  centers  of  lo  cm.  (3.9  in.)  under  the  crown  had 
taken  place,  the  deflection  of  the  arch  down  to  its  final  position  was  only  17  mm. 
(0.669  ii^O  5  the  amount  of  the  deflection  of  the  arch  measured  to  the  center- 
ing was  not  uniform.  This  observed  deflection  measured  to  the  centering 
amounted  on  the  right  span  to  6.5  mm.  (0.256  in.),  and  on  the  left  one  to  10  mm. 
(0.394  in.).    Before  and  after  the  lowering,  the  width  of  the   hinge  opening 


Schnin  a-b.-1:20. 


Fig.  304. — Thoroughfare  construction  between  the  piers  on  the  left  bank. 


between  the  stones  was  measured,  but  a  diminution  of  not  more  than  j\  mm. 
(0.004  ii^O  could  be  observed.  Shifting  of  the  foundations  could  not  be  ascer- 
tained with  certainty  with  the  instrumental  arrangements  at  hand. 

The  computations  for  the  deflection  of  the  crown  gave  a  satisfactory  agreement 
with  the  measured  amount,  but  also  gave  the  conclusion  that  small  deviations 
from  the  profile  planned  influenced  the  deflection  considerably.* 

The  hinge  openings  of  the  arch  were  filled  with  cement  mortar,  so  as  to  pro- 
tect the  steel  hinges  from  rust.  The  mobility  of  the  joint  was  maintained  by  a 
layer  of  asphalt,  concreted  into  the  center  of  the  opening. 

Since  the  actual  cost  of  the  bridge  was  only  about  260,000  M.  ($62,000  approx.), 
it  is  demonstrated,  as  far  as  the  Griinwald-Isar  bridge  is  concerned,  that  arched 
bridges  with  a  proper  arrangement  of  reinforced  concrete,  and  of  large  spans, 
can  compete  successfully  with  steel  construction. 


*  See  the  description  in  the  "  Schweizerischer  Bauzeitung,"  1904. 


ARCHES 


281 


In  Fig.  305  is  shown  a  smaller  bridge  with  a  hingeless  arch  and  similar  super- 
structure. The  span  is  only  23  m.  (75.4  ft.),  the  thickness  of  the  arch  at  the 
crown  being  35  cm.  (13.7  in.),  and  at  the  springings  60  cm.  (23.6  in.). 

Besides  the  usual  round  rod  reinforcement,  plate  or  lattice  girders  are  of 
advantage  for  the  reinforcement  of  arch  bridges,  according  to  the  Melan  system. 
A  description  follows  of  the  highway  bridge  over  the  Mosel  Road  at  Wasser- 
liesch,  in  which  a  shallow  depth  of  construction  was  secured  by  arranging  the 
reinforcement  in  the  form  of  lattice  girders. 

In  this  bridge,  which  replaced  a  grade  crossing,  not  enough  space  existed 
between  the  clearance  required  and  the  arch  profile  to  erect  a  scaffold  of  the  usual 
variety;  even  an  iron  structure  to  support  the  sheeting  would  commonly  occupy 
the  whole  of  the  clearance  space,  which  cannot  usually  be  spared.    Such  metal 


Fig.  305. — Nagold  Bridge  near  Cain  (Wiirtemberg). 


forms  can  only  be  employed  to  advantage  when  they  may  be  used  a  large  number 
of  times  in  a  more  uniform  structure. 

Since  an  arch  of  the  Monier  variety  was  in  this  case  out  of  the  question,  it 
was  necessary  to  build  the  reinforcement  so  that  it  formed  a  supporting  arch 
structure,  upon  which  the  forms  could  be  hung  in  such  manner  that  the  space 
below  the  arch  would  be  entirely  free  from  scaffolding. 

In  Fig.  306  these  steel  supporting  members  are  shown  in  detail.  There  are 
six  members  side  by  side  in  a  distance  of  0.9  m.  (2.95  ft.),  and  they  are  secured 
against  lateral  tipping  by  a  light  horizontal  connection.  At  the  grades  of  the 
upper  and  lower  layers  was  a  network  of  longitudinal  and  cross  wires  7  mm. 

in.)  in  diameter,  so  as  better  to  tie  together  the  concrete. 

The  3.5  cm.  (1.38  in.)  form  boards  were  supported  by  50X50X7  mm.  (2X 
2X1Q  in.  approx.)  angle  irons,  bent  into  a  curve,  and  hung  from  the  arch  rein- 
forcement by  15  mm.        in.  approx.)  screw  bolts  at  intervals  of  80  to  100  cm. 


282 


CONCRETE-STEEL  CONSTRUCTION 


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(30  to  40  in.).  When  the  forms  were  removed,  the  bolts  were  withdraw^n  from 
the  concrete  matrix,  and  the  holes  so  left  were  filled  with  cement  mortar. 

The  steel  arch-supports  with  fixed  ends  consisted  of  four  angles  50X50X7 
mm.  (2X2X1%  in.  approx.),  which  were  held  together  at  distances  of  50  cm. 
(19.7  in.)  by  plates.  The  ends  were  supported  in  their  exact  position  by  pairs 
of  wedges.  The  calculation  of  the  arch  supports  was  made  on  the  assumption 
that  half  the  arch  would  be  placed  at  one  time,  although  this  unfavorable  load- 
ing could  be  obviated  by  commencing  the  concreting  at  both  springings  and 
at  the  crown  at  the  same  time. 

In  the  foregoing  type  of  construction,  the  concrete  is  loaded  much  less 
than  in  the  usual  reinforced  arch,  because  the  dead  load  is  carried  exclusively 
by  the  arch-supports,  and  the  weight  of  the  additional  superstructure  and  of  the 
live  load  is  carried  jointly  by  the  concrete  and  the  steel  in  proportion  to  their 
elastic  deformations. 

A  greater  advantage  of  the  above  described  construction  is  that  the  removal 
of  the  forms  can  be  done  earlier  (after  about  eight  days)  and  without  special  care, 
whenever  the  concrete  has  become  so  hard  that  it  can  be  left  free  between  the 
steel  ribs  with  safety.  In  the  usual  reinforced  concrete  arch  the  scaffolding 
should  not  be  removed  inside  of  about  four  weeks. 

A  bridge  of  the  Melan  type  with  three  hinges  is  described  with  all  details  in 
"  Beton  und  Eisen,"  No.  Ill,  1903.  In  very  large  spans  the  girders,  later  to  be 
concreted,  give  a  guaranty  for  the  maintenance  of  the  proper  form  of  the  arch, 
which  is  of  great  importance,  which  can  be  secured  with  w^ooden  centers  only 
through  considerable  care.  The  girders  always  carry  the  whole  or  a  greater 
part  of  the  centering.  When  a  wooden  scaffold  is  employed  in  addition,  it  can 
be  made  much  hghter  than  is  otherwise  necessary.  A  noticeable  appHcation  of 
the  Melan  system  was  also  made  in  the  Chauderon-Montbenon  bridge  in  Laus- 
anne, where  the  high  supporting  scaffolding  was  omitted. 

Newer  Methods  of  Arch  Construction  in  Reinforced  Concrete. — In  the 
foregoing,  it  was  always  assumed  that  the  carrying  part  of  the  arch  was 
rectangular  in  section.  The  serviceable  application  of  reinforced  concrete  to 
arch-like  structures  of  other  shapes  has  lately  been  made.   There  may  be  mentioned: 

(a)  Reinforced  arches  of  rectangular  section,  to  which  the  throroughfare 
is  attached  by  hanging  columns.  In  this  case  the  thoroughfare  can  also  act  as 
a  tie  so  that  all  horizontal  shear  is  taken  from  the  abutments.  (See  ''Deutsche 
Bauzeitung,  Zementbilage,"  Nos.  17  and  21,  1905).  The  best  example  of  this  kind 
is  the  railroad  bridge  over  the  Rhone  at  Chippis,  with  a  span  of  60  m.  (197  ft.), 
in  which  the  thoroughfare  contains  an  expansion  joint  at  the  center  of  the  span. 
(See  "  Schweizerisch  Bauzeitung,"  1907.) 

{b)  According  to  a  method  of  construction  already  much  used  in  Switzer- 
land, by  Maillart  of  Zurich,  for  reinforced  concrete  arch  bridges,  the  side  walls 
and  the  deck,  which  both  consist  of  reinforced  concrete,  can  be  built  in  cantilever 
form  decreasing  in  depth  from  the  abutments  to  the  crown.  Since  the  combina- 
tion of  the  several  parts  of  this  section  is  perfect,  it  can  be  employed  in  its  entirety 
for  carrying  stresses.  This  construction  appears  to  be  specially  adapted  only 
for  three-hinged  arches.    See  " Schweizerische  Bauzeitung,"  October  i,  1904. 

(c)  The  arch  can  also  be  constructed  cf  separate  ribs  of  rectangular  section 


284 


CONCRETE-STEEL  CONSTRUCTION 


placed  side  by  side,  wherein  spirally  reinforced  concrete  affords  extra  strength. 
The  thoroughfare  is  then  supported  by  columns  from  the  arch  ribs.  They  are 
prevented  from  moving  laterally  by  bulkheads  or  continuous  slabs.  If  the 
latter  are  so  built  as  to  be  flush  with  the  upper  layers  of  the  ribs  at  the  crown 
and  with  the  lower  layers  near  the  abutments,  this  arrangement  provides  con- 
siderable resistance  against  deformations  from  normal  stresses  and  from  change 
of  temperature. 

{d)  Spirally  reinforced  concrete  is  especially  applicable  to  bridges  and  to 
arches  consisting  of  separate  ribs  of  octagonal  or  rectangular  section,  and  also  to 
truss-like  members  in  the  form  of  beams  or  arches.  Banded  concrete  is,  however, 
not  yet  well  known,  so  that  designers  do  hardly  more  than  experiment  with  it. 
See  Considere,  Essai  a  outrance  du  Pont  dTvry,"  "  Annales  des  Fonts  et  Chaus- 
sees,"  No.  3,  1903,  and  "  Beton  und  Eisen,"  No.  i,  1904. 

RESERVOIRS. 

The  Monier  system,  with  its  network  of  wire,  is  well  adapted  to  the  con- 
struction of  reservoirs  of  all  kinds.  When  a  serviceable  method  of  computing 
slabs  was  found,  the  thickness  of  walls  and  amount  of  reinforcement  was 
determined  with  a  proper  margin  of  safety.  Even  earlier,  an  extensive  apphcation 
of  reinforced  concrete  was  made  in  the  construction  of  various  reservoirs  for 
industrial  purposes.  Because  of  the  satisfactory  experience  with  these  structures, 
reinforced  concrete  now  occupies  an  even  broader  field  in  this  line. 

In  the  catalogue  issued  by  Wayss  and  Freytag,  in  1895,  are  found  several 
examples  of  this  apphcation,  explained  by  sketches.  There  may  be  mentioned: 
for  paper  mills, — bleaching  cyHnders,  drip  boards,  chloride  holders,  acid  tanks, 
mixing  vats,  settling  basins;  for  breweries, — barley-soaking  vats,  drying  arches, 
icehouses;  for  tanneries, — tan  pits,  etc.;  for  pulp  mills, — similar  parts  to  those  in 
a  paper  mill,  and  so  on. 

If  the  tanks  are  circular,  the  horizontal  reinforcement  has  to  resist  the  tangen- 
tial stress,  the  vertical  rods  acting  only  as  ties.  Long  walls  of  rectangular  tanks 
are  rigidly  connected  to  the  bottom,  which  is  always  constructed  as  a  reinforced 
slab  monolithic  with  the  walls.  In  that  case  the  largest  stresses  are  in  the  vertical 
reinforcement.  The  water  tightness  is  obtained  by  a  water-proof  cement  coating 
on  the  inside. 

The  cylinders  are  protected  from  the  injurious  influence  of  acids  by  a  cover- 
ing of  porcelain  tile.  The  construction  of  bleaching  cylinders  with  brick  or  mass 
concrete  walls  has  proven  unsatisfactory,  since  the  heating  through  of  the  relatively 
thick  wall  takes  too  long.  When  the  hot  paper  stock  is  introduced,  the  inside  is 
warm  and  the  outside  cold,  and  cracking  takes  place.  Cylinders  of  reinforced 
concrete  do  not  possess  this  disadvantage,  since  with  the  thinner  walls,  a  quicker 
equalization  of  temperature  results,  and  furthermore,  the  reinforcement  resists 
the  stresses  produced. 

Several  constructions  adapted  to  the  needs  of  special  industries  will  be  described 
in  detail.  The  largest  water  supply  reservoirs  may  be  built  of  reinfored  concrete 
in  various  ways.    Either  the  walls  may  be  made  of  mass  concrete  or  masonry, 


RESERVOIRS 


285 


and  reinforced  concrete  used  for  roof  and  partitions;  or  bottom,  walls  and  top 
may  all  be  built  of  reinforced  concrete. 

An  example  of  the  first  kind,  which  is  most  often  employed,  is  shown  in  Fig. 
307.  The  top,  when  built  in  reinforced  concrete,  offers  economic  advantages 
over  the  usual  arched  construction  only  when  the  price  of  gravel  and  its  transporta- 
tion cost  is  high. 


Fig.  307. — Water  reservoir  with  reinforced  concrete  cover. 


Reservoirs  entirely  of  reinforced  concrete,  up  to  about  300  cu.m.  (79,250  gals.) 
capacity,  are  usually  of  hemispherical  or  cylindrical  form,  with  a  dome-shaped 
top.  Fig.  308  shows  a  section  through  a  hemispherical  reservoir,  such  as  are 
commonly  builj:  for  small  water  sup- 
plies. With  graphical  methods  of 
calculating  domes,  the  various  stresses 
to  resist  loads  in  the  directions  of 
the  meridians  and  parallels  may  be 
found,  and  the  reinforcement  corres- 
pondingly determined. 

Cylindrical  reservoirs  with  flat 
dome-shaped  tops  must  be  provided 
with  a  heavy  tension  ring  to  resist 
the  horizontal  shear  of  the  dome  and 
transfer  it  to  the  cylinder.  Fig.  310  Fig.  308. — Water  reservoir  in  hemispherical  form, 
shows  a  water  reservoir  7  m.  (22.9  ft.) 

in  diameter,  reinforced  with  round  rods  and  having  I-beams  as  an  inclosing 
ring,  and  with  a  top  constructed  to  resist  heavy  street  traffic. 

To  construct  of  reinforced  concrete  the  whole  of  a  large  reservoir  of  elongated 
rectangular  plan  is  not  always  as  economical  as  a  well  built  reservoir  of  mass  con- 
crete. Under  certain  circumstances  the  former  allows,  however,  a  better 
employment  of  the  ground  area  available  and  is  to  be  recommended  with  ex- 
pensive gravel  and  sand.  In  Fig.  311  is  shown  such  a  reservoir,  of  4000  cu.m. 
(1,057,000  gals.)  capacity,  for  Brussels.  The  side  walls,  which  withstand  a  water 
head  of  2  m.  (6.6  ft.)  were  reinforced  concrete  12  cm.  (4.7  in.)  thick,  spanning 
between  the  top  and  the  bottom.  The  beams  in  the  top  form  rectangular  panels, 
so  that  the  deck  slab  was  reinforced  in  two  directions. 

Water  towers  are  well  adapted  for  construction  in  reinforced  concrete,  the 
substructure  as  well  as  the  tank  being  of  the  same  material.    The  structural 


286 


CONCRETE-STEEL  CONSTRUCTION 


members  may  be  placed  on  the  outside  and  thus  add  to  the  architectural  ap- 
pearance. 

Figs.  312-313  show  a  cyHndrical  tank  with  dome-shaped  bottom,  supported 
on  masonry  walls.  The  shear  produced  by  the  arched  form  of  the  bottom  was 
resisted  by  eight  round  rods  40  mm.  (i^  in.)  in  diameter,  forming  a  ring. 


Fig.  309. — Cylindrical  water  reservoir  with  dome-shaped  top. 


J'lacheiserv 


In  Fig.  314  is  shown  the  reservoir  of  the  Gross wartenberg  water  supply  system. 
The  tank  overhangs  the  supports,  because  of  the  interior  construction.  The 
insulating  wall  is  also  constructed  of  reinforced  concrete.  The  details  of  construc- 
tion, and  all  the  arrangements  are  shown  in  the  figure. 

Figs.  315-316  show  a  vertical  section  and  general  view  of  the  water  tower 

in  Rixensart,  in  which  the  substructure 
consists  of  a  pleasing  reinforced  concrete 
frame,  the  panels  of  which  are  filled  with 
brickwork. 

As  far  as  the  tensile  stresses  involved 
are  concerned,  gas  receiver  frames  can 
profitably  be  constructed  of  reinforced 
concrete.  For  the  upright  guides,  special 
concrete  piers  must  be  built,  both  out- 
side and  in  connection  with  the  receiver 
walls.    See  Fig.  317. 

In  the  cylindrical  walls  of  such  a  tank, 
besides  the  circumferential  stresses,  bend- 
ing stresses  in  a  vertical  direction  exist, 
developed  because  the  cylindrical  walls  are 
prevented  by  their  rigid  connection  with 
the  bottom  from  assuming  the  deforma- 
tion corresponding  to  the  circumferential 
stresses,  that  is,  of  increasing*  in  di- 
ameter. Since  the  cylinder  is  prevented  to  the  greatest  extent  from  enlarging 
near  the  base,  vertical  tensile  stresses  appear  there  on  the  inner  side,  while  they 
act  near  the  top  on  the  outer  side.    These  vertical  stresses  are  cared  for  by 


Fig.  310.  —  Cylindrical 
water  reservoir.  Ten- 
sion ring  of  the  dome- 
shaped  top  and  con- 
nection with  the  cylin- 
drical sides. 


RESERVOIRS 


287 


vertical  steel  near  the  inner  and  outer  faces  of  the  cylindrical  wall,  which  must  be 
strongly  connected  with  the  bottom.    Concerning  an  investigation  into  the  method 


Fig.  311. — Water  reservoir  of  4000       (141,275  ft^)  capacity  for  Brussels.  Cross-section. 


of  calculating  these  bending  stresses  by  Reich,  see  "  Beton  und  Eisen,  1907, 
No.  10. 

Silos. — Silos  are  bins  for  certain  dry  materials,  such  as  grain,  coal,  cement, 
ore,  broken  stone,  etc.,  in  which,  because  of  their  shaft-like  arrangement,  the 


material  which  was  received  above,  can  be  extracted  when  necessary  from  the 
lowest  point  of  the  bin.    In  this  connection  may  be  distinguished  large  silos 


288 


CONCRETE-STEEL  CONSTRUCTION 


I^iG-  313- — Cylindrical  tank  with  dome-shaped  bottom.    Detail  of  reinforcement. 


RESERVOIRS  .  289 


Fig.  314. — Grosswartenberg  water  tower. 


290 


CONCRETE-STEEL  CONSTRUCTION 


without  separate  partitions,  or  such  as  are  relatively  large  in  area  with  respect 
to  their  height;  and  cellular  silos,  or  silos  consisting  of  compartments  of  rectangular,, 
or  better,  of  square,  round,  or  hexagonal  section. 

Examples  of  the  first  variety,  without  interior  division  walls,  are  the  ore  bins 
for  the  Burbach  smelters,  the  general  arrangement  and  details  of  which  are 
given  in  Figs.  318  to  320.  The  foundation  of  these  bins  consists  of  a  continuous 
reinforced  concrete  slab  70  cm.  (27.6  in.)  thick,  which  distributes  the  load  uni- 
formly upon  the  soil  at  a  stress  of  1.5  kg/cm^  (1.5  tons/ft^).  Since  the  columns, 
were  spaced  3.33  m.  (10.9  ft.)  apart  in  both  directions,  reinforced  concrete  beams- 


Fig.  315. — Water  tower  in  Rixensart.  Flg.  316. — Water  tower  in  Rixensart. 


were  built  into  the  foundation  slab  in  each  direction  under  the  rows  of  columns^ 
the  square  panels  between  being  correspondingly  reinforced.  The  60X60  cm. 
(24X24  in.)  openings  in  the  funnel-shaped  square  panels  in  the  bottoms  of  the 
bins  were  supplied  with  slide  valves.  Beams  also  extend  in  both  directions 
over  the  rows  of  columns.  The  outside  walls  6  m.  ^23. 6  ft.)  high,  were  anchored 
to  the  beams  in  the  bottoms  of  the  bins  by  ribs  25  cm.  (9.8  in.)  thick.  Between 
them  the  outer  wall  acts  as  a  continuous  reinforced  concrete  slab.  The  top  of  the 
wall  is  stiffened  by  a  somewhat  thicker  rib.  Three  raikoad  trestles  enter  the  bin 
on  25  cm.  (9.8  in.)  thick  supporting  walls,  6.66  m.  (21.8  ft.)  apart,  carried  by  the 
lower  columns.    The  trestle  stringers  are  continuous  reinforced  concrete  girders. 

Spaced  26.4  m.  (87.2  ft.)  apart  are  expansion  joints  through  floors  and  walls.. 
All  exposed  edges  of  ribs,  supporting  walls  and  columns  are  protected  against 
wear  by  channel  and  angle  irons. 


SILOS 


291 


A  smaller  silo,  without  interior  partitions,  is  shown  in  section  and  plan  in  Fig. 
319.  The  coal  in  storage  can  be  discharged  directly  into  the  boiler  room  through 
several  openings.  The  walls  carry  the  lateral  pressure  of  the  stored  material 
horizontally  to  the  columns  which  are  tied  together  by  the  roof,  and  between  the 
funnels  by  cross-beams,  45X45  cm.  (17.7  in.)  in  section. 

In  the  silo  for  the  Odenwalder  copper  wwks  (Fig.  321)  a  trestle  of  the  kind 
described  above  is  built  over  the  several  pockets.    The  latter  are  of  an  elongated 


Fig.  317. — Gas  holder  of  the  Jagstfeld  Railroad  Station. 


rectangular  form,  and  make  a  row  along  the  length  of  the  building  serving  for 
the  storage  of  crushed  porphyry  of  various  sizes  of  particles.  The  building  is 
described  at  greater  length  by  the  author  in  Beton  und  Eisen,"  No.  i,  1903, 
with  details,  to  which  reference  should  be  made. 

The  malt  silo  of  the  Lowen  brewery  in  Munich,  contains  cells  3.5X3.75  m. 
(11. 5X12. 3  ft.)  in  plan,  16.5  m.  (54  ft.)  high,  with  a  capacity  of  2200  hi.  (6242 
bushels).  The  points  of  intersection  of  the  walls  are  supported  by  reinforced 
concrete  columns,  which  carry  the  load  to  a  continuous  foundation  slab,  i  m. 
(39.4  in.)  thick,  so  that  the  foundation  pressure  is  only  2.5  kg/cm^  (2.5  tons/ft^). 


292  CONCRETE-STEEL  CONSTRUCTION 

The  outer  walls  show  pleasing  reinforced  concrete  ribs,  the  panels  between  which 
are  filled  with  brickwork.    Between  the  latter  and  the  outside  silo  walls  an 


Fig.  318. — Section  and  plan  of  the  Burbach  ore  bins. 


insulating  air  space  is  left.  The  silos  are  emptied  by  screw  conveyors  located 
at  the  level  of  the  bottoms  of  the  bins,  in  the  lateral  passage  (Fig.  322). 

While  in  usual  structures  the  silo  funnels  are  constructed  simply  as  hanging 
pyramids,  the  large  ones  in  the  saw-dust  bins  of  the  pulp  mill  shown  in  Fig.  323 


SILOS 


293 


are  supported  by  a  T-beam  deck,  and  stiffening  walls  are  erected  upon  this. 
The  outside  walls  8  m.  (26.2  ft.)  apart  are  also  tied  together  by  an  anchor  beam 
in  the  center  of  each  panel  between  the  cross  walls.  The  principal  reinforcement 
of  the  outside  walls  runs  vertically  between  the  two  horiontal  wall  beams.  Fig. 
324  gives  a  view  of  these  silos,  which  are  erected  on  a  high  brick  substructure. 
Arched  beams  are  used  for  the  roof  girders. 


k  tp9  4 


Horizontal-section. 
Fig.  319. — Coal  pocket  in  Kirn. 


Examples  of  cellular  silos  with  very  large  pockets  of  6.7X8.5  m.  (21.9X27.9 
ft.)  ground  plan,  are  those  of  Figs.  325  and  326,  showing  the  coal  pockets  of  the 
Volklingen  smelters.  In  this  case  it  was  essential  that  the  underside  of  the  floor 
should  be  absolutely  flat,  making  it  necessary  that  the  beams  and  girders  extend 
above  the  slab.  The  silo  rested  on  a  continuous  reinforced  concrete  foundation 
slab.    Details  of  the  girders  are  shown  in  Fig.  326. 

Reinforced  concrete  can  doubtless  be  considered  the  best  building  material 
for  silos;  since,  aside  from  its  fire-proof  qualities,  the  reinforcement  of  the  walls 
affords  at  the  same  time  the  best  anchorage  for  them.  It  so  happens  most 
advantageously  that  both  functions  of  the  horizontal  reinforcement  in  the  cell 


SILOS 


295 


walls  are  not  necessary  at  the  same  time;  the  anchorage  stresses  being  greatest 
with  simultaneous  filling  of  adjacent  pockets,  when,  however,  the  bending  stresses 
are  nill;  and  with  maximum  bending  stresses,  as  when  a  single  j)ocket  is  filled,  the 
anchorage  stresses  diminish  one-half.  As  to  the  bending  with  axial  tension, 
which  occurs  in  this  case,  see  page  127.  The  constructive  excellence  of  reinforced 
concrete  has  further  been  proved,  because,  during  the  past  year,  in  the  construc- 
tion of  a  large  silo  foundation,  an  additional  application  has  been  found.  The 
following  is  a  single  example  of  silo  construction  executed  by  Wayss  and  Freytag 
this  year  and  last. 


Fig.  321. — Reinforced  concrete  silo  for  the  Odenwald  Copper  Works  at  Rossdorf. 


Fig.  329  is  a  view  of  the  reinforcement  in  the  funnel  of  the  silos  of  seven 
hexagonal  pockets  16  m.  (52.4  ft.)  high,  for  the  Alsen  Portland  cement  works 
in  Itzehoe. 

Since  the  walls  of  square  cells  are  to  be  considered  as  fully  restrained  at  the 
corners  when  adjacent  pockets  are  filled,  the  moment  at  the  centers  of  panels 

is  — ,  and  in  the  corners  — .    For  this  reason  the  walls  should  be  twice  as  thick 
24  12 

at  the  corners  as  at  the  centers.    In  Fig.  327  is  shown  the  reinforcement  of  the 

walls  for  a  silo  with  44  square  cells  of  4X4  m.  (13. i  ft.)  ground  plan  for  the 

Alsen  Portland  Cement  Works  at  Itzehoe.    It  is  seen  that  the  wall  thickness  is 

doubled  at  the  corners.    The  outside  walls  are  covered  with  brickwork,  with  an 

insulating  air  space  between  it  and  the  reinforced  concrete  walls,  and  its  support 


'296 


CONCRETE-STEEL  CONSTRUCTION 


Fig.  322. — Lower  Brewery,  Silo,  Munich.       Fig.  324. — Sawdust  bins  of  the  Waldhof  Pulp  Mill. 
Half  longitudinal  section 


Cross-section.  Longitudinal  section. 


Fig.  323. — Sawdust  bins  of  the  Waldhof  Pulp  Mill. 


SILOS 


297 


SO  01 


t 


VSO 


085 


i 


I 


080 


0/J 


XJL 


HI 


oes  065 


Fig.  325. — Valklingen  coal  pockets. 


Fig.  326. — Valklingen  coal  pockets.    Details  of  the  rods  in  the  girders  in  the  bottoms. 


298  CONCRETE-STEEL  CONSTRUCTION 

is  obtained  at  certain  levels  from  projecting  reinforced  concrete  ribs  monolithic 
with  the  concrete  wall. 

A  view  of  two  silos,  each  with  seven  hexagonal  cells,  is  shown  in  Fig.  328. 
The  bending  moments  in  them  are  relatively  less  than  in  square  pockets. 

Smelters  require  a  special  variety  of  silo,  called  an  ore  pocket,  the  cells  of  which 


are  not  arranged  in  two  or  more  separately  entered  groups.  In  Figs.  330  and  331 
are  given  a  cross-section  and  a  longitudinal  section  of  such  an  ore  pocket  for  the 
Mosel  smelter  at  Maiziere,  which  is  178  m.  (584  ft.)  long.  The  sloping  floor 
slabs  were  supported  by  very  heavy  cross-beams,  upon  which  also  rested  the 
reinforced  concrete  columns  supporting  the  three  railroad  trestles.  Because  of 
the  great  length  of  the  construction,  four  expansion  joints  were  installed.  For 


Fig.  329. — Cement  bins.    Reinforcement  of  the  funnels  near  the  hexagonal  cell  bottoms. 


300 


CONCRETE-STEEL  CONSTRUCTION 


Fig.  331.- 


Longitudinal  section. 


SILOS 


301 


especially  designed  to  resist  shearing  stresses,  special  bent  rods  being  employed, 
while  the  upper  and  lower  rods  were  not  diverted. 

In  the  lower  part,  the  longitudinal  walls  spanned  directly  from  one  girder  to 


SILOS 


303 


the  next,  while  in  the  upper,  open  part  tney  extended  between  vertical  reinforcing 


beams  which  were  anchored  together 
by  tie-beams  at  the  tops.  The  column 
arrangement  is  not  regular,  being 
made  to  conform  to  the  requirements 
of  the  necessary  lateral  passages 
which  required  a  i)eculiar  modification 
from  that  used  in  the  usual  sections 
of  silos. 

The  emptying  of  the  pockets  takes 
place  at  the  lowest  point  and  also  at 
the  center,  through  properly  con- 
structed gates  which  were  concreted 
into  the  outside  walls  and  the  sloping 
bottoms  by  angle  iron  frames.  Three 
parallel  railroad  tracks  ran  over  the 
whole  length  of  the  bins,  two  of  i 
m.  (39.37  in.)  gauge  from  the  mines, 
and  one  of  standard  gauge  for  the 
transportation  of  coke. 

The  extent  of  this  plant  required 
for  its  construction  about  500  t.  (550 
tons)  of  fabricated  round  rods.  The 
time  of  construction  was  somewhat 
more  than  six  months  from  the  time 
of  starting  the  foundations.  In  Fig. 
334  is  show^n  the  reinforcement  of 
the  lateral  beams,  and  in  Fig.  335  that 
of  the  sloping  bin  bottoms. 

The  ore  pockets  in  Dudelingen 
have  a  capacity  of  about  5000  cu.m. 
(176,500  cu.ft).  The  heavy  cross 
walls  which  carry  the  sloping  bottoms 
and  the  longitudinal  walls,  are  rein- 
forced and  constructed  as  somewhat 
overhanging. 

The  beams  of  the  ore  crusher 
floor  are  stiffened  against  the  heavy 
vibrations  of  traffic  by  special  cross- 
beams. The  roof,  which  is  supported 
by  free  columns  7.3  m.  (24  ft.)  high,  is 
arched,  and  is  supplied  w^ith  tension 
members.  To  resist  the  wind  pressure 
in  a  longitudinal  direction  along  the 
roof,  diagonal  members  are  supplied. 


In  this  case,  also,  several  expansion  joints  were  installed,  to  effect  which  the 
necessary  columns  and  cross  walls  were  built  double. 


304 


CONCRETE-STEEL  CONSTRUCTION 


Fig.  336  shows  a  cross  and  a  longitudinal  section  of  the  building,  and  Fig. 
337  shows  the  placing  of  the  reinforcement. 


Fig.  336. — Diidelingen  ore  pockets.    Cross  and  longitudinal  sections. 


The  Getreide  silo  at  Hafen  in  Genoa  is  shown  in  Fig.  338.  The  building 
consists  in  the  main  of  an  enlargement  of  the  silos  previously  built  by  Hennebique. 


Fig.  337. — Diidelingen  ore  pockets.    Placing  reinforcement. 

Besides  a  large  granary,  divided  into  rooms,  a  new  building  was  erected  65 
m.  (203  ft.)  long,  40  m.  (131  ft.)  wide,  and  30  m.  (98  ft.)  high,  which  contains 


SILOS 


305 


126  bins,  3X3  m.  (9.8  ft.),  and  3X5  m.  (9.8X16.4  ft.)  in  plan.  These  increased 
the  capacity  from  about  28,000  t.  (30,800  tons)  to  50,000  t.  (55,000  tons). 

The  whole  building  rests  on  reinforced  concrete  columns  90X90  cm.  (35.4 
in.)  thick,  which  have  a  carrying  capacity  of  400  t.  (440  tons).  Under  the  silos 
are  five  loading  tracks,  the  substructure  for  which,  together  with  the  loading 
platform,  are  of  reinforced  concrete.  The  total  load  of  the  silo  and  tracks  was 
uniformly  distributed  by  means  of  a  continuous  reinforced  concrete  slab  upon 
^  twenty -year-old  fill  of  an  inkt  from  the  harbor,  in  such  manner  that  the  maxi- 
mum soil  presure  was  only  1.7  kg/cm^  (1.7  tons/ft^). 

The  construction  of  the  slab  with  quasi-inverted  arches  was  decided  upon 
because  of  the  necessity  of  having  a  flat  top  surface,  so  as  to  secure  enough  space 
for  the  loading  tracks  and  platforms. 


Fig.  338. — Getreide  grain  isilo,  Genoa.  Cross-section. 


Furthermore,  this  arrangement  enabled  the  loading  tracks  to  be  brought  close 
to  the  rigidly  set  automatic  scales  connected  to  the  mouths  of  the  bottom  funnels 
produced  by  concreted  inclined  surfaces  on  top  of  the  horizontal  bin  bottoms. 
The  full  load  of  the  bottom  is  hung  upon  the  cross  walls,  which  are  correspond- 
ingly reinforced  so  as  to  form,  in  combination  with  the  bottom  and  floor  slabs, 
a  continuous  beam  15  m.  (49.2  ft.)  deep,  with  three  spans  each  8  m.  (26.2  ft.)  long. 

The  construction  of  the  large  floor,  which  was  divided  into  rooms,  is  clearly 
shown  in  the  section,  w^hile  the  succeeding  illustrations  give  pictures  of  the  method 
of  erection  of  the  building. 

During  the  period  of  construction  of  200  working  days,  the  following  quan- 
tities of  material  were  used : 

About  900  t.  (990  tons)  steel  almost  entirely  of  German  manufacture. 
About  2,900,000  kg.  (6,393,400  lbs.)  cement  from  the  mills  of  Flli.  Palli 

Caroni  Deaglio  Casale. 


306 


CONCRETE-STEEL  CONSTRUCTION 


About  ii,ooo  cu.m.  (14,380  cu.yds.)  sand,  gravel  and  fine  slag,  largely 
from  the  shores  of  both  rivers. 

The  concrete  was  mixed  by  an  electric  impulse  mixer  with  fixed  drum. 

In  the  best  months,  from  August  to  November,  when  the  daily  rate  was 
about  60  cu.m.  (78  cu.yds.)  of  finished  concrete,  the  monthly  performance  averaged 
about  200,000  lire  ($38,600). 

It  is  very  important,  in  the  design  of  silos,  to  know  the  lateral  pressure  exerted 
against  the  walls  by  the  material  in  the  pockets.  In  silos  of  large  size,  without 
cross  walls,  or  those  with  very  long  rectangular  cells,  the  computation  is  to  be 


Fig.  339. — Getreide  grain  silo,  Genoa.    Reinforcement  of  the  bottoms. 


made  according  to  the  usual  formulas  for  earth  pressure.  Neglecting  the  friction 
against  the  walls,  the  total  lateral  pressure  on  the  height  h  is 

P  =  ir/^2tan2(45°-^/2), 

and  the  pressure  on  a  differential  area  at  the  height  h  is 


=     tan2(45° -0/2). 


For  several  materials  the  quantities  in  Table  XXXVII  can  be  employed: 


Material.  kg/m.^  lbs/It.^  kg/m^  lbs/ft' 

Gas  coal   800-900  50-56  45  146  30 

Cement   1400  87  40  305  62 

Small  slag   1600-1800  100-112  45  290  59 

Malt  -  530  33  22  240  49 

Wheat   820  51  25  333  68 

Minette  (ore)  .  .  2000  125  45  343  70 


Fig.  341. — ^View  of  the  silo  in  Genoa  with  the  colonnade  toward  the  harbor, 


308 


CONCRETE-STEEL  CONSTRUCTION 


In  celled  silos  of  considerable  height,  these  figures  give  very  heavy  pressures; 
in  the  lower  parts,  and  the  lightening  effect  of  the  friction  of  the  material  against 
the  walls  may  be  considered.  Two  pubHcations  exist  agreeing  in  all  essentials 
concerning  the  computation  of  the  lateral  pressures  in  silo  cells,  by  Janssen  in  "  Zeits- 
chrift  des  Vereins  deutscher  Ingenieure,"  1895,  p.  1046,  and  by  Konen  in  the 
"  Zentralblatt  der  Bauverwaltung,"  1896  p.  446.  The  friction  between  the  mate- 
rial and  the  side  acts  so  that  the  lateral  pressures  can  never  exceed  a  certain  frac- 
tion of  /'max.-  This  fraction  may  be  introduced  and  the  weight  of  any  layer  will 
then  become  a  function,  in  the  computation  of  the  frictional  resistance  developed. 


Fig.  342. 

In  Fig.  342  let  it  be  assumed  that  in  a  full  cell,  at  a  depth  a  layer  of 
thickness  d  x\'$>  cut  out,  then  the  following  forces  are  active.* 

qF^  from  above,  where  q  is  the  special  pressure  in  a  vertical  direction; 

Fydx^  the  weight  of  the  layer; 

(q-\-dq)  F,     the  vertical  resistance  from  below; 
pUdx,  the  horizontal  pressure  against  an  area  Udx; 

pU  t2in(j)idx,  the  frictional  resistance  of  the  walls  at  this  horizontal  pressure 
and  acting  upward; 

where  F  is  the  area  of  the  cell  (Trans.); 
and  U  is  the  circumference  of  the  cell  (Trans.). 


From  the  equating  of  the  vertical  components  of  the  opposite  forces,  it  follows 


that 


dq=dx(  y—p^t2in 


In  a  material  devoid  of  cohesion  under  a  vertical  pressure  q,  there  is  developed 
a  lateral  pressure. 

tan2(45-^/2), 


whence 


dq=dx(^-q  tan2[45° -9^)/ 2]  ~ tan  . 


*  See  Konen  "  Zentralblatt  der  Bauverwaltung,"  1896. 


SILOS 


309 


If  ?n  is  inserted  in  place  of  the  constant  factor  tan^  (45°— <y^)/2)  ^tan  then 

wiU 

dq^dx(j'-qm), 

or 

dy  =  , 

j  —  mq 

from  which,  by  integration, 

x  =  — ^  log  {Y  —  mq)-\-C. 

7)1 

Since  q  must  equal  o,  for  x  =  o,  the  value  of  the  constant  of  integration  is 


so  that 


or 


y-7nq 
w^c=iog  , 


^  —  mq  I 


Finally  there  results 


tan2(45-(/)/2). 


The  presures  p  and  5  are  thus  seen  to  vary  with  the  depth  x,  and  with 

.    U  . 

rease  of  the  ratio  — ,  sin 
r 

for  x  =  oc  and  at  that  point 


increase  of  the  ratio        since  m  then  varies.    They  reach  their  greatest  values 
r 


^     tan2  (45-^/2)  J  tan  9^)1 


/'max  — 


—  tan  01 

The  last  result  can  be  immediately  deduced  on  the  assumption  that  the  maximum 
pressure  exists  when  the  frictional  resistance  on  the  wall  is  equal  to  the  total  weight 
of  any  layer,  or  that 

pmax  Udx  i2in(j)i==  Fydx, 

from  which 


J  tan<;6i 


310 


CONCRETE-STEEL  CONSTRUCTION 


With  square  section  of  pocket  of  side  5, 


Pmax 


rs 


4  tan  01* 


The  computation  of  special  values  by  the  formula  above  given  is  not  readily 
accomplished,  and  the  following  simpler  method  may  be  followed:  At  the  top 
the  limiting  value  of  the  lateral  pressure  is  to  be  computed  by  the  formula  p=-fh 
tan^  (45°— 0/2),  while  at  a  certain  depth  the  value  reaches  that  given  by  the  formula 


U 


tan  (hi 


after  which  it  remains  constant.  It  may  be  assumed  then,  in  accordance  w^ith 
Fig.  343,  that  the  area  representing  the  lateral  pressure  is  bounded  by  two  straight 

lines.  One  line  starts  at  the  origin  of  the 
true  curve,  and  is  tangent  to  it  at  that 
point,  while  the  other  is  the  asymptote 
of  the  curve.  This  simple  method  of 
computation  also  gives  greater  security, 
which  is  the  more  advisable,  since  experi- 
ments have  shown  that  while  in  motion, 
materials  often  exert  greater  pressures  than 
when  at  rest. 

The  greatest  bottom  pressure  can  also  be 
computed,  so  that 


tan2(45-0/2)' 


Fig 


343. — Diagram  of  assumed  lateral 
pressures  on  silo  walls. 


When  still  more  safety  is  desired,  the 
weight  of  the  whole  cell  contents  can  be 
assumed  as  taken  by  the  bottom.  Even 
with  small  depth  the  actual  weight  will  be   less   than  the   greatest  bottom 
pressure  A^,  since  this  value,  like  ^^ax.'  really  exists  only  at  infinitely  great 
depths. 

Tan  9^1  represents  the  coefficient  of  friction  between  the  contained  material 
and  the  cell  walls,  which  may  be  assumed  at  J  to  |  for  grain,  but  never  greater 
than  tan  0,  or  than  the  coefficient  of  friction  of  the  material  upon  itself.  Unplas- 
tered  reinforced  concrete  walls  usually  give  greater  values  of  tan  that  is,  smaller 
lateral  pressures  than  wooden  walls,  and  are  thus  also  advantageous  in  this 
respect. 

When  the  plan  of  the  pockets  deviates  from  the  square  form,  the  bending 


pl2  pl2 

moments  cannot  be  computed  with  the  formulas  —  and  — . 

12  24 


If  the  form  is  a 


SILOS 


311 


rectangle  of  length  /  and  breadth  b,  there  results,  by  the  law  of  virtual  displace- 
ments, or  with  the  formulas  of  continuity  of  beams,  at  the  corner  (corner  moment), 


M=  .  p  .  - — r  • 

12     ^  /+6 


In  the  center  of  the  side  /  the  moment  is 


while  in  the  center  of  the  side  b  it  is 


The  formulas  are  deduced  upon  the  assum])tion  that  both  walls  are  of  equal 
strength,  and  the  deformation  of  the  supports  provided  by  the  cell  walls  is 
negligible. 

Experiments  concerning  the  actual  wall  and  bottom  pressures  in  silos  have 
been  published  by  Prante,  in  the  Zeitschrift  des  Vereins  deutscher  Ingenieure," 
1896,  p.  1 1 22,  and  by  Pleizner,  in  the  same  publication  in  1906,  Nos.  25  and  26. 
The  Pleiszner  experiments  are  especially  valuable,  since  they  were  very  extensive 
and  were  made  on  silos  of  usual  dimensions.  Various  kinds  of  grains  were  em- 
ployed as  filling  material,  and  because  of  the  restrained  conditions  of  the  cells 
it  was  possible  to  increase  the  weight  of  the  wheat  (per  hektoliter)  from  790  kg. 
to  846  kg/m^  (49-3  to  52.8  Ibs/'ft^).  Bottom  and  side  pressures  were  determined 
by  the  exact  measurement  of  deflections  produced  by  wooden  shutters  after  other 
methods  had  proved  unsatisfactory. 

The  Pleiszner  measurements  gave  a  very  satisfactory  confirmation  of  the 
theory  developed  and  the  accompanying  formulas  for  p  and  q.  When  the 
corresponding  values  of  the  angles  of  friction  0  and  0i,  were  computed  from 
the  measured  values  of  p  and  q  by  this  formula,  the  results  were  obtained 
with  wheat  in  timber  bins  of  2.51  to  2.90  m.  (8.2  to  9.5  ft.)  constant  cell  section. 


Table  XXXIX 


Measured  Pressure. 

Depth  X. 

Computed  Angles. 

P 

Q. 

m. 

ft. 

kg/m^ 

Ibs/ft2 

lbs  ft2 

2-7 

8.9 

500 

102 

1610 

31°  40' 

28°  20' 

5-4 

17.7 

740 

2490 

510 

32°  40' 

28°  20' 

8.1 

26.5 

910 

186 

3100 

635 

33°  00' 

27°  00' 

The  values  of  cj)  and  ^1  thus  remain  nearly  constant,  as  is  assumed  in  the 
formula,  or  in  other  words:  with  exactly  chosen  values  of  ^  and  0i  the  formulas 
for  p  and  q  give  equivalent  results.  Similar  results  were  also  secured  with  other 
silos.    These  apply  primarily  only  to  quiescent  loading,  since  the  invariably 


312 


CONCRETE-STEEL  CONSTRUCTION 


untrustworthy  measurements  showed  variations  in  the  pressure,  when  the  exit 
gates  were  opened.  In  some  cases  an  increase  to  one  and  a  half  times  the  static 
pressure  was  found.  Account  of  this  increase  can  be  secured  by  introducing 
smaller  angles  cf)  and  into  the  computations.  On  the  other  hand,  an  additional 
factor  of  safety  for  the  walls  is  given,  in  that  the  given  pressure  exists  at  the 
centers  of  the  side  walls,  while  it  is  somewhat  less  against  the  corners,  although 
an  equal  distribution  was  assumed  in  the  computations.  If  calculations  are  made 
with  the  values  of  ^  =  25°,  tan  9^1=0.3,  given  on  p.  306  for  wheat,  there  are 
obtained  for  the  large     timber  silo,"  the  results  of  Table  XL. 


Table  XL 


Depth. 

P 

Q 

Computed. 

Observed. 

Computed. 

Observed. 

m. 

2-7  

kg/m^ 
720 
I160 
1430 

kg/m2 
500 
740 

kg/m2 
1770 
2860 
3520 

kg  'm2 
1610 
2490 
3100 

5-4  

8.1  

By  the  exact  formula  for  p,  a  security  of  about  ij  is  thus  secured,  which  is 
sufficiently  exact  in  the  light  of  the  increase  of  the  side  pressure  during  empty- 
ing of  the  pocket.  This  is  especially  true  when  the  earth  pressure  formula  is 
used  for  the  upper  part  and  simply  ^^ax.  lower. 


FURTHER  EXAMPLES  OF  THE  USE  OF  REINFORCED 

CONCRETE 

Tunnel  at  Wasserburg  a.  Inn. — This  tunnel  pierces  a  17  m.  (55  ft.)  high 
street  embankment,  and  cuts  it  at  a  rather  sharp  angle.  The  reinforcement  of 
the  tunnel  was  in  the  form  of  an  angle  and  channel  lattice  work  riveted  together, 
that  served  at  the  same  time  as  a  support  for  the  forms  which  were  temporarily 
bolted  thereto,  and  as  a  reinforcement  of  the  tunnel  lining  in  place  of  the  round 
rods  commonly  employed.  Over  the  ironwork  used  for  the  form  support,  was 
driven  a  4  cm.  (1.6  in.)  sheeting,  and  the  tunnel  profile  was  executed  from  meter 
to  meter,  with  perpendicular,  projecting  forms  which  were  stiffened  from  the 
starting  point.*  The  metal  frame  was  cross-braced  with  timber,  so  as  to  be  able 
to  withstand  the  heavy  and  one-sided  pressure  of  the  earth.  After  the  concreting 
and  hardening  of  each  ring,  the  wooden  interior  forms  were  shifted. 

Shipping  Platforms  of  Reinforced  Concrete. — The  shipping  platform  of  the 
Strasburg-Neudorf  Railroad,  shown  in  Fig.  345,  is  constructed  on  the  principle 
of  a  retaining  wall  like  an  L.  The  vertical  wall  spans  the  space  between  the 
ribs,  which  are  anchored  in  the  ground  slab.  The  latter  are  arched  upward,  so 
that  they  can  carry  the  load  of  the  superimposed  earth  to  the  ribs  without  the 


*  Poling  boards. — (Trans.) 


EXAMPLES  OF  THE  USE  OF  REINFORCED  CONCRETE  313 

developing  of  bending  stresses.  At  distances  of  about  lo  m.  (32.8  ft.)  expansion 
joints  are  provided  wtiich  cut  the  entire  section.  , 


Fig.  344. — Tunnel  for  the  Royal  Bavarian  City  Railroad  of  Wasserburg  a.  Inn. 


Fig.  345. — Loading  platform  of  the  Strassburg-NeUclort  Railway  Station. 


Cooling  Towers  of  Reinforced  Concrete. — Reinforced  concrete  made  possible 
cooling  tanks  of  larger  diameter  than  the  usual  ones  constructed  of  wood.  Aside 
from  the  question  of  durability,  they  have  the  advantage  that  all  bracing  may  be 


314 


CONCRETE-STEEL  CONSTRUCTION 


omitted.  The  tower,  35  m.  (115  ft.)  high,  erected  for  the  Differdingens  Steel 
Works  (Figs.  346  and  347),  consisted  of  a  cyhndrical  substructure  of  16  m.  (42  ft.) 
diameter,  upon  which  was  a  dome,  the  top  of  which  carried  a  cylinder  7  m. 
(23  ft.)  in  diameter.  The  construction  was  designed  only  for  its  own  weight 
and  wind  pressure. 

Pipes  of  Reinforced  Concrete  are  excellent,  even  for  those  which  must  with- 
stand interior  pressures.    They  are  reinforced  against  this  pressure  by  bands 


Fig.  346. — Cooling  tower  of  the  Differdingen  Steel  Works. 


applied  to  the  cement  mortar,  which  is  also  reinforced.  External  forces  are  also 
readily  resisted  by  these  reinforced  concrete  pipes,  even  when  thin.  Reinforced 
pipes  are  especially  appHcable  for  turbine  penstocks,  and  have  been  built  for 
pressures  up  to  20  m.  (66  ft.)  head  of  water.  For  a  whole  year  some  experimental 
pipes  made  by  Wayss  and  Freytag  were  exposed  in  their  Munich  storage  yard 
to  a  pressure  of  three  atmospheres,  and  even  at  a  computed  stress  in  the  wire 
reinforcement  of  1700  to  1800  kg/cm^  (24,179  to  25,560  lbs/in^)  no  cracks 
appeared,  and  the  pipes  were  water-tight.    The  permissible  stress  in  the  reinforce- 


EXAMPLES  OF  THE  USE  OF  REINFORCED  CONCRETE  315 


ment  of  pipes  subjected  to  internal  pressure  should  not  exceed  600  kg/cm^  (8534 
lbs/in2). 

Further  mention  of  applications  of  reinforced  concrete  will  be  omitted  without 
pretention  to  having  covered  com})letcly  its  field.  A  wide  region  which  it  has 
scarcely  as  yet  entered,  is  in  the 
building  of  dykes  and  dams,  in 
which  the  new  type  of  construction 
will  provide  more  secure  and  more 
economical  structures. 

In  the  calculation  of  structures 
in  reinforced  concrete,  the  relation- 
ship between  the  external  forces 
and  the  desired  security  is  not  as 
readily  determined  as  in  steel  con- 
struction. In  this  connection  it 
is  necessary  not  only  to  carry 
through  computations  with  regard 
to  all  possible  relationships  within 
the  project  itself,  but  also  to  take 
into  consideration  the  possibility  of 
a  loosening  of  the  reinforcement 
from  moisture,  and  to  take  this  into 
account  in  preparing  designs.  Re- 
inforced concrete  constructions 
which  are  to  be  built  in  accord- 
ance with  fundamental  statical 
conditions,  demand  of  the  de- 
signing engineers,  that  sufficient 
knowledge  and  experience  are 
applied  to  secure  the  best  arrange- 
ment in  each  special  case. 

Furthermore,  the  execution  of 
reinforced  concrete  work  requires 
intimate  knowledge  and  great  care, 
which  cannot  be  secured  from 
every  contractor.  It  is  in  a  certain 
sense  a  matter  of  trust,  deserved 
only  by  special  firms.  The  worst 
accidents  of  the  past  year  also 
show  clearly  that  the  execution  of 
any  work  should  not  be  separated 
from  its  design.  Only  when 
both  are  combined  in  an  under- 
taking can  the  right  comprehension  be  secured  of  the  statical  relations  in  the 
construction,  and  it  will  be  more  necessary,  the  more  the  construction  deviates 
from  the  standard  designs  of  any  "  system." 

It  should  be  striven  at  all  times  to  place  constructions  on  a  scientific  basis,  and 


347- 


-Reinforced  concrete  cooling  tower  of  the 
Differdingen  Steel  Works. 


316 


CONCRETE-STEEL  CONSTRUCTION 


not  have  the  ambition  to  clothe  them  in  a  given  "  system,"  in  the  clear  knowledge 
that  every  such  system  "  hinders  further  development.  And  further,  the  more 
or  less  extensive  protection  afforded  by  a  patent  in  various  countries  is  to  be 
considered.  The  fewer  patents  of  non-essential  details  that  are  granted  (as  is  the 
case  in  Germany),  the  quicker  will  the  general  and  scientific  point  of  view  secure 
attention. 

In  the  next  few  years,  the  several  scientific  commissions  investigating  reinforced 
concrete  in  various  countries  will  provide  much  additional  experimental  material, 
which  will  shed  much  light  on  the  yet  unsettled  questions.  In  Germany,  besides 
the  active  "  Eisenbeton-kommission  der  Jubilaumsstiftung  der  Deutscher  Industrie," 
there  has  of  late  also  been  added  in  this  direction  the  "  Deutscher  Ausschuss  fiir 
Eisenbeton  in  Tatigkeit,"  which  is  receiving  ample  funds  for  investigation  from 
the  government.  In  its  programme,  the  solution  of  important  questions  is  pro- 
vided for.  Through  the  joint  work  of  several  testing  laboratories  it  is  practically 
guaranteed  that  results  will  not  be  awaited  for  any  great  length  of  time. 


APPENDIX 


PRELmiNARY  RECOMMENDATIONS  (LEITSATZE)  FOR  THE 
DESIGN,  CONSTRUCTION,  AND  TESTING  OF  REIN- 
FORCED  CONCRETE  STRUCTURES 

(Prepared  by  the  Verband  Deutscher  Architekten-  und  Ingenieur  Vereine  and  the  Deutscher 
Beton  Verein  in  the  year  1904.) 

I.  GENERAL 

These  recommendations  relate  to  buildings  or  structural  members  composed 
of  concrete  with  steel  reinforcement  of  any  desired  variety,  in  which  both  elements 
of  construction  attain  common  static  efficiency  in  supporting  the  load.* 

II.  BUILDING  PRELIMINARIES 

As  a  rule,  the  form  of  contract  for  reinforced  concrete  structures  should  include 
the  following  points: 

1.  Plans,  showing  the  design  as  a  whole  and  in  detail. 

2.  Static  computations,  giving  the  contemplated  loads  and  proving  compre 
hensively  the  adequate  safety  of  the  structure. 

3.  Specifications,  concerning  the  origin,  character,  and  composition  of  the  mate- 
rials intended  for  use. 

4.  Specifications,  as  to  the  tensile  strength  of  the  reinforcement  and  of  the 
compressive  strength  (in  cubes)  of  the  concrete. 

5.  Explanations,  of  difficult  construction  features,  of  the  rate  of  execution,  etc. 

These  contract  papers  should  be  signed  by  their  author,  and  before  the  com- 
mencement of  the  work  by  the  contractor  who  is  undertaking  the  erection  of  the 
reinforced  concrete  structure. 

Special  dispensations  granted  by  local  authorities  as  to  methods  of  construction, 
in  nowise  relieve  the  contractor  from  full  responsibility  as  to  design  and  execution. 

*  The  Leitsatze  thus  apply  equally  to  stone  construction  supplied  with  reinforcement  in 
such  manner  that  the  steel  embedded  in  the  mortar  resists  the  tensile  or  bending  stresses. 

317 


318 


CONCRETE-STEEL  CONSTRUCTION 


III.  CHECKING  PLANS 

Since,  at  the  present  time,  no  generally  recognized  theory  exists  for  the  design 
of  reinforced  concrete  structures,  it  is  recommended  that,  for  the  present,  all  plans 
for  reinforced  concrete  buildings  be  checked  by  the  approximate  methods  con- 
tained in  the  appendix,  and  there  illustrated  by  examples. 

IV.  BUILDING  CONSTRUCTION 

A,  SUPERVISION  OF  WORK  AND  EMPLOYEES 

The  contractor  for  reinforced  concrete  work  must  entrust  the  construction 
only  to  such  persons  as  are  thoroughly  familiar  with  that  method  of  building. 

For  the  actual  work,  trained  workmen  must  be  employed,  under  the  constant 
supervision  of  technical  superiors  or  experienced  foremen  familiar  with  this  method 
of  construction. 

If  requested  so  to  do  by  the  builder,  or  authorized  officials,  the  contractor 
must  produce  proof  that  those  entrusted  with  the  direction  and  supervision  of  the 
work  are  reliable  persons  who  have  had  previous  successful  experience  in  the 
execution  of  reinforced  concrete  work. 

B.  MATERIALS  AND  THEIR  HANDLING 

1.  Reinforcement 

Before  use,  the  steel  is  to  be  cleaned  from  all  dirt  and  grease,  as  well  as  from 
loose  rust. 

It  is  recommended  that  reinforcing  rods  subject  to  tension  should  be  hooked 
or  otherwise  so  formed  as  to  make  more  difficult  the  slipping  of  the  steel  in  the 
concrete. 

Welding  is  to  be  avoided  as  much  as  possible,  and  the  weld  is  never  to  be  located 
at  a  dangerous  point. 

The  placing  of  the  reinforcement  must  be  so  done  that  it  shall  occupy  as 
exactly  as  possible  the  positions  shown  on  the  plans,  and  care  must  be  taken  that 
it  is  completely  embedded  in  the  concrete. 

The  protection  of  the  reinforcing  rods,  that  is,  the  distance  of  the  surface  of  the 
steel  from  the  outside  of  the  concrete,  should  not  be  less  than  i  cm.  (0.4  in.)  as  a 
rule.  With  rods  of  less  diameter  than  i  cm.,  the  covering  may  be  reduced  to  0.5 
cm.  (0.2  in.)  if  a  coat  of  plaster  is  to  be  apphed  later. 

2.  Cement 

Only  cement  of  recognized  good  quality,  equal  to  standard  Portland  cement, 
is  to  be  used. 


APPENDIX 


319 


3.  Sand,  GravcU  and  Otlier  Aggregates 

Sand,  gravel,  and  other  aggregates  must  he  suital)le  for  making  concrete 
(see  II,  Jt,  and  \,A  4). 

The  size  of  particles  of  the  aggregate  must  be  such  as  to  allow  satisfactory 
working  of  the  concrete  between  the  reinforcing  bars  and  between  them  and 
the  forms. 

Acid  *  slag  may  be  used  as  aggregate  only  when  its  harmless  character  has 
been  demonstrated. 

Jf.  Concrete 

As  a  rule  the  concrete  must  develop  after  hardening  for  28  days  under 
normal  climatic  conditions,  in  30  cm.  (11. 8  in.)  cubes,  a  compressive  strength 
of  from  180  to  200  kg/cm-  (2560  to  2845  lbs  in-). 

It  must  be  so  wet  that  perfect  contact  and  covering  l)y  the  mortar  of  the 
concrete  will  be  secured  with  the  reinforcement. 

The  mortar  contained  in  any  concrete,  wherein  is  used  a  sand  with  grains 
of  varying  size  up  to  7  mm.  {\  in.  approx.),  must  not  be  leaner  than  1:3.  Aggre- 
gate of  gravel  or  hard  broken  stone  of  proper  size  may  be  used  in  quantity  up 
to  that  of  the  sand. 

The  preparation  of  the  concrete  must  be  so  conducted  that  the  quantities 
of  the  several  ingredients  are  constantly  under  control.  When  mixing  concrete 
according  to  volume  (i.e.,  by  measure),  it  is  to  be  understood  that  the  cement 
is  to  be  placed  in  the  measure  without  falling  (i.e.,  being  shaken  in). 

In  translating  parts  by  volume  into  parts  by  weight,  it  is  to  be  understood 
that  a  cubic  meter  of  Portland  cement  weighs  1400  kg.  (87.4  lbs/ft-"^). 

C.  FORMS  AND  SUPPORTS.    TIME  OF  REMOVAL 

The  forms  must  be  so  strong,  and  so  well  fastened  and  supported  that  an  exact 
reproduction  of  the  intended  structural  part  will  be  secured.  They  must  also 
withstand  the  tamping  of  the  concrete  in  thin  layers,  and  must  admit  of  ready 
removal  without  danger,  allowing  necessary  supports  to  remain  in  place. 

The  period  which  must  elapse  between  the  final  depositing  of  the  concrete 
and  its  uncovering  (i.e.,  the  removal  of  forms  and  supports)  depends  upon  the 
prevaihng  weather  conditions,  the  spacing  of  supports,  and  the  weight  of  the 
structural  parts  in  question.  The  side  forms  of  beams  and  their  sup])orts,  as 
well  as  the  forms  for  floor  slabs  of  small  span,  may  be  removed  as  soon  as  the 
concrete  has  hardened  sufficiently — that  is,  in  a  few  days — while  the  supports 
under  the  beams  should  not  be  removed  in  less  than  14  days.  With  large  spac- 
ing between  supports,  and  with  structural  parts  of  considerable  sectional  area, 
a  period  of  from  4  to  6  w^eks  should  elapse,  in  some  cases. 

In  buildings  of  several  stories  the  supports  of  the  lower  floors  should  not  be 


*  As  a  rule  testing  with  litmus  paper  will  suffice. 


320 


CONCRETE-STEEL  CONSTRUCTION 


removed  until  the  slabs  have  so  hardened  that  their  carrying  strength  is  sufficient 
to  support  the  superimposed  load. 

If  frost  should  occur  during  the  period  of  hardening,  the  time  of  removal  should 
be  extended  at  least  as  long  as  the  period  of  frost. 

D.  PROTECTION  OF  STRUCTURAL  PARTS 

As  soon  as  completely  tamped,  all  reinforced  concrete  work  must  be  protected 
in  a  suitable  manner  against  injury,  and  from  external  influences  which  might 
have  a  detrimental  effect  upon  the  attainment  of  proper  carrying  power.  Care 
must  also  be  taken  that  after  a  structure  has  attained  proper  strength,  it  is  not 
weakened  by  any  proceeding  such  as  the  cutting  of  holes  or  slots  for  pipes,  etc., 
at  improper  points. 

V.  INSPECTION  AND  TEST  OF  WORK. 

A.  TESTS  DURING  ERECTION 
As  a  rule,  the  tests  must  cover: 

1.  The  adequacy  of  constrution  of  forms  and  supports. 

2.  The  accuracy  of  the  use,  arrangement,  and  size  of  reinforcement  accord- 
ing to  the  drawings. 

3.  The  employment  of  the  proper  concrete  mixture. 

4.  The  determination  that  the  materials  employed  possess  the  strength  de- 
manded by  the  designer.  (See  II,  4-)  This  determination  can  be  made  either 
by  testing  30  cm.  (11.8  in.)  cubes  in  a  press  at  the  building  site,  for  the  manu- 
facture of  which,  concrete  mixed  at  the  building  is  to  be  used,  or  by  securing  a 
certificate  of  test  from  a  testing  laboratory,  concerning  the  strength  of  portions 
of  the  material  taken  from  the  building. 

Under  certain  circumstances  the  test  may  be  conducted  by  constructing  a 
test  member  (such  as  a  T-beam),  and  after  a  28-day  period  of  hardening,  load 
it  to  failure,  noting  the  deflection  as  exactly  as  possible. 

B.  TESTS  AFTER  COMPLETION 
These  tests  should  include: 

1.  The  determination  whether  the  structural  members  have  properly  har- 
dened before  the  forms  are  removed. 

2.  The  determination  whether  all  structural  members,  upon  being  uncovered, 
are  free  from  defect. 

3.  The  determination  whether  the  calculated  strength  of  the  parts  of  the 
construction  has  actually  been  attained,  by  cutting  (i.e.,  by  making  several 
holes  in  various  floors). 

4.  Under  certain  circumstances,  the  making  of  load  tests.  Such  tests  are 
always  to  be  undertaken  when  there  is  reason  to  beheve  the  structural  parts  are 


APPENDIX 


321 


defective  in  construction,  or  that  they  have  been  injuriously  affected  as  to  their 
supporting  power,  by  some  cause. 

Load  tests  should  be  undertaken  only  after  the  concrete  has  hardened  for 
45  days. 

In  making  load  tests  of  slabs  and  beams,  if  g  is  the  dead  weight,  and  p  the  uni- 
formly distributed  working  load,  with  a  whole  panel  loaded,  when  the  working 
load  is  not  greater  than  looo  kg/m^  (205  lbs/ft^),  the  total  test  load  should  not 
exceed  0.8  ^  +  1.8  p. 

When  the  working  load  is  greater  than  1000  kg/m-  (205  lbs/ft-),  the  test  load 
is  to  be  proportionately  decreased. 

Structural  members  thus  loaded  may  be  considered  as  sufficiently  safe  when 
no  noticeable  permanent  deformation  takes  place. 

It  is  important  to  determine  as  accurately  as  possible  the  deflection  at  dif- 
ferent stages  of  the  load  test. 

C.  DUTIES  OF  THE  CONTRACTOR 

The  contractor  must  be  attentive,  and  when  so  requested  by  the  builder,  or 
the  proper  officials,  must  furnish  proof  of  the  correctness  of  his  plans  and  the 
excellence  of  his  work,  in  accordance  with  sections  V,  A  4,^,  B  3  and  V,  B  4- 

The  expense  of  such  plans  should  bear  a  proper  relation  to  the  total  cost  of 
the  building. 

VI.  EXCEPTIONS 

Departures  from  the  foregoing  recommendations  are  permissible  when  they 
appear  to  be  warranted  by  exhaustive  experiment,  by  experience  gained  in  the 
erection  of  the  building,  or  in  the  opinion  of  competent  judges. 


APPENDIX  TO  THE  FOREGOING  RECOMMENDATIONS  RE- 
GARDING METHODS  OF  CALCULATION  TO  BE  USED  IN 
TESTING  REINFORCED  CONCRETE  STRUCTURES 

I.  FUNDAMENTAL  ASSUMPTIONS 
A.  EXTERNAL  FORCES 
1.  Loads 

A  distinction  must  be  made  between 

(a)  the  dead  weight  of  the  reinforced  concrete,  which  is  to  be  computed  at 
2400  kg/m^,  unless  a  less  weight  can  be  demonstrated. 
{b)  Other  constant  loads. 
(c)  The  live  or  moving  load. 


322 


CONCRETE-STEEL  CONSTRUCTION 


2,  Reactions,  Moments,  Shears 

(a)  For  the  computation  of  reactions,  moments,  and  shearing  forces,  the  rules 
of  statics  and  of  the  science  of  elasticity  are  to  be  determinative. 

(b)  To  determine  the  ultimate  strength,  the  most  unfavorable  distribution 
or  location  of  the  live  or  moving  loads  is  to  be  considered. 

(r)  Possible  impact  effects  may  be  considered  by  the  customary  increase  of 
the  moving  load. 

(d)  The  distance  between  points  of  support  is  to  be  computed, 

1.  In  beams,  as  the  center  to  center  distance  between  supports. 
Except  when  the  calculation  is  governed  by  other  considerations; 

2.  In  freely  supported  floor  slabs,  as  the  free  span  of  the  slabs  plus  the  thick- 
ness of  the  slab  at  its  center; 

3.  In  continuous  floor  slabs,  as  the  distance  fi?om  center  to  center  of  beams. 

(e)  If  conditions  at  supports  produce  restraint  and  continuity  of  slabs  and 
beams,  the  bending  moments  which  appear  at  those  points  must  have  reinforce- 
ment placed  near  the  upper  stressed  surface  in  proportion  to  the  bearing  area. 

If  a  continuous  beam  or  slab  cannot  be  so  computed,  or  in  regard  to  the 
latter,  if  no  restraint  is  certain  at  a  beam  or  wall,  then,  with  equal  panels  and 

uniformly  distributed  load,  the  moment  is  not  to  be  taken  less  than  ~-  over  the 

8 

supports  or  than  —  at  the  centers  of  panels.    With  unequal  panels  ~  is  to  be 

considered  the  moment  of  the  largest  panel. 

Restraint  of  beams  at  wafls  exists  in  few  instances  and  should  therefore  be 
ignored  unless  special  structural  arrangements  make  the  restraint  certain.  In 
these  cases  the  restraint  is  to  be  demonstrated  by  calculation. 

(/)  In  designing  supports  the  possibility  of  eccentricity  of  loading  must  be 
considered. 


B.  INTERNAL  FORCES 

{a)  The  internal  stresses  and  strains  in  the  concrete  are  to  be  determined 
upon  the  assumption  of  its  being  a  homogeneous  material.  The  modulus  of 
elasticity  of  concrete  in  compression  will  be  assumed  as  constant,  so  that  the 
ratio  of  the  modulus  of  elasticity  of  steel  to  that  of  concrete  will  be  E^'M^^n^i^, 
and  accordingly,  the  steel  section  is  to  be  employed  in  computations  at  15  times, 
its  actual  value. 

{h)  The  determination  of  the  internal  tensile  stesses  and  strains  in  the  steel 
is  to  be  made  on  the  assumption  that  the  steel  carries  the  whole  tensile  stress,  the 
tensile  strength  of  the  concrete  being  ignored. 

{c)  Steel  in  compression  is  to  be  computed  at  15  times  its  area.  The  risk 
of  buckling  is  to  be  considered. 


APPENDIX 


323 


C.  SAFE  STRESSES 

{a)  The  permissible  stress  depends  upon  the  ultimate  strength  of  the  mate- 
rial and  the  method  of  com])utation. 

{b)  Upon  the  assum]:)ti()n  that  the  concrete  when  28  days  old  has  a  compres- 
sive strength  of  180  to  200  kg/ cm-  (2560  to  2845  lbs/in-),  and  the  steel  a  tensile 
strength  of  3800  to  4000  kg  cm-  (54,041  to  56,893  lbs/ in-),  the  following  stresses 
should  not  be  exceeded  when  computed  by  the  approximate  formulas  given 
later: 

Concrete  in  compression  from  flexure,      40  kg/cm^  (569  lbs/in^); 

direct  compression,  35  (498      "  ); 

shear  from  flexure,  4-5*"      (64      "  ); 

adhesion,  7.5  (107      "  ); 

Steel  in  tension,  1000  (14,223  ). 

For  concrete  of  higher  compressive  strength,  correspondingly  higher  per- 
missible stresses  may  be  allowed,  up  to  50  kg/cm^  (711  lbs/ in-).  Similar  allowance 
may  be  made  for  steel  of  higher  tensile  strength. 


II.  APPROXIMATE  METHODS  OF  COMPUTATION 


A.  SIMPLE  BENDING 

1.  Rectangular  Sections.  Slabs. 

(a)  With  single  reinforcement.  Let 
Fe  =  the  section  of  the  reinforcement  in  square  centimetersf  found  in  a 
breadth  b  (in  centimeters)  of  the  slab. 
h  =  the  height  above  center  of  reinforcement. 

Ec 

AI  =  moment  of  the  external  forces,  in  centimeter-kilograms. 
F  =  the  shearing  forces  in  corresponding  sections,  in  kilograms.  Then, 
according  to  Fig.  i ,  the  distance  of  the  neutral  plane  below  the  top  is 


nFf 


\         2bll  1 


*  Wherever,  in  T-beams  or  girders,  there  exists  a  shearing  stress  greater  than  the  permissible 
one  of  4.5  kg/cm^  (64  lbs/in^),  a  part  of  the  lower  reinforcing  rods  may  be  bent  obliquely 
upward  and  anchored  in  the  zone  of  compression  in  such  manner  as  to  resist  the  diagonal  tensile 
stresses  produced  at  an  angle  of  45°  in  the  neighborhood  of  the  points  of  support,  and  which 
may  be  considered  equivalent  to  the  shearing  stresses.  The  number  of  rods  thus  to  be  bent  is  to  be 
determined  by  the  amount  of  diagonal  tensile  stress  in  excess  of  4.5  kg/cm^  (64  lbs/in^), 
which  they  must  carry. 

It  is  recommended  that  in  T-beams  a  rounded  or  inclined  transition  be  made  from  the  ribs 
to  the  floor  slabs  so  as  better  to  transfer  the  shearing  forces  from  one  to  the  other. 

t  The  formulas  are  equally  serviceable  when  inches  and  pounds  are  substituted. — (Trans.) 


324 


CONCRETE-STEEL  CONSTRUCTION 

2M 


The  stress  in  the  concrete  is  ob- 


the  stress  in  the  steel  is 


the  shearing  stress  is 


Ge- 


^0- 


bx{h—x/T,) 
M 

"Fe{h-xisy 

V 

"b{h-xisy 


the  adhesive  stress  on  the  reinforcement,  which  must  be  considered  at  certain 
sections,  is 

^  bro  

Circumference  of  the  reinforcement* 

As  a  rule,  the  calculation  of  the  shearing  and  adhesive  stresses  in  slabs  is 
unnecessary. 


 T  

 » — \ 

-t  -  > 

^  A 

1  ' 
t  1 

Y 

z  ^ 


 K_ 

"}<  \ — ~ 

< 

jyc  > 

A 

\  ^ 

Fig.  2. 

B 

Fig. 


(&)  With  double  reinforcement.  According  to  the  nomenclature  of  Fig.  2, 
the  distance  x  of  the  neutral  plane  may  be  determined  by  the  quadratic  equation : 

x^  +  2xn — ,  ^-rihFe  +  h'Fe)\ 

0  0 

X  having  been  ascertained,  the  compressive  stress  in  the  concrete  will  be 

6Mx 


Ob- 


bx^iS^i  -  x)  ^  6F/n{x  -  h'){h-h') 
the  tensile  stress  in  the  lower  reinforcement  is 

ob{h—x)n 

<7e  =  ; 

X 

and  the  compressive  stress  in  the  upper  reinforcement  is 

,    Gb{x  —  h')n 


X  / 


APPENDIX 


325 


2.  T-Beams 


The  effective  width  of  slab  h  is  to  be  taken  as  h  ^  J/,  in  which  /  represents  the 
distance  between  the  supports  of  the  beam,  but  h  must  not  be  greater  than  the 
beam  spacing. 

Tw^o  different  cases  exist: 

{a)     x^d  (see  Fig.  3). 

X   i   )• 


Fig.  3a  and  3^. 


The  formulas  given  under  A,  1  a  apply  in  this  case.  Under  certain  circum- 
stances the  shear  in  the  rib  and  the  adhesive  stress  on  the  reinforcement  at  the 
support  must  be  calculated.    These  are 


-Co 


boih-x/sY 


a 

i  \ 

A 

Circumference  of  the  reinforcement 
{b)     x>d    (see  Fig.  4). 

Ignoring  the  small  compressive  stress  in  the  rib,  there  are  found 
2nhFe-{-hd'^ 


J/ 


Fig.  4. 


x  = 


(7e  = 


2{nFe  +  hd) 
M 


and 


and 


d  d^ 

y  ^=   .  4"  

2  6{2x—dy 


Ob- 


OeX 


Fe{h—x-\-y)       ^"  n{h-x)' 

B.  COMPRESSION 

The  reinforcement  of  columns  must  aggregate  at  least  0.8%  of  the  total  area. 
The  reinforcement  subject  to  stress  is  to  be  secured  against  buckling  by  lateral 
ties  (usually  of  round  iron).  The  spacing  of  these  ties  should  not  exceed  the 
diameter  of  the  column. 


I.  SUPPORTS  FOR  SAFE  LOADS 


{a)  Central  Loading. — If  Fb  represents  the  area  of  the  concrete  member,  the 
permissible  load  will  be 

P  =  ob{Fb-^nFe), 


326  CONCRETE-STEEL  CONSTRUCTION 

where  ^  =  15.  Further, 


Fb+nFe  „  ,  I'd 

re-]  • 

n 


(b)  Eccentric  Loading  (Bending  with  Axial  Stress). — The  computation  can 
be  made  in  the  same  manner  as  for  a  section  of  homogeneous  material,  provided 
that  in  all  expressions  representing  the  areas  of  sections  and  their  moments  of 
inertia,  the  area  of  reinforcement  is  to  be  computed  at  ^  =  15  times  its  actual 
value  compared  with  the  concrete  section.  If  tension  occurs,  the  reinforcement 
located  on  the  tension  side  must  be  capable  of  carrying  it. 

No  danger  of  buckling  exists,  provided  the  supports  have  at  least  the  following 
dimensions: 


Stress  of  the  Concrete, 
kg 'cm 2              1              lbs /in2 

Least  Diameter  of  Round 
Column  in  Terms  of 
its  Length. 

Least  Length  of  Short 
Side  of  Rectangular 
Column  in  Terms  of 
its  Length. 

427 

1/18 

1/  21 

35 

498 

1/20 

40 

569 

1/16 

1/19 

45 

640 

1/15 

1/18 

50 

711 

I./14 

1/17 

Since  few  experiments  exist  concerning  the  resistance  to  buckling,  smaller 
dimensions  than  those  given  above  should  not  be  used. 


IIL  EXAMPLES  OF  THE  METHOD  OF  COMPUTATION  FOR  A 

FEW  SIMPLE  CASES 


A.  SIMPLE  BENDING 
1.  Slabs 

(a)  Freely  Supported  Slabs  with  Single  Reinforcement. 

Clear  span   2.00  m. 

Thickness  of  slab   o-i5  " 


Distance  between  supports   2.15m. 

The  live  load  />  =  iooo  kg/m^,  the  dead  load  ^  =  0.15X2400  =  360  kg.,  and 
thus  the  total  load  ^  =  1360  kg/m^,  and  the  moment  for  i  m.  breadth  (see  Figs. 
I  and  5), 

M  =  i36oX^^^^-Xioo  =  78,583  cm..-kg. 


APPENDIX 


327 


In  a  breadth  of  i  m.  were  9  lower  rods  of  10  mm.  diameter,  making  F^.  =  'j.oy 
cm2  For  ^  =  13.5,  ^  =  15,  and  6  =  100,  the  distance  x  of  the  neutral  axis  below 
the  top  of  the  slab  is 


cm. 


fi 

y>7 

1 

,  ^1 

mmm 

  J^Jj§^^^$J^ 

1 

1 

Stress  in  the  concrete 
2M 


Fig.  s. 


2X78,583 


bx{Ii-xls)    iooX4-39(i3-5 -4-39/3) 
Stress  in  the  steel 


29.7  kg/cm^. 


M 


78,583 


7:^-^923  kg/cm2. 


Fe{h-x/s)  7.07(13.5-4.39/3) 
The  shear  at  the  support  is  7  =  ^X1360X2.0=1360  kg.,  so  that  the  shear- 


ing stress  is 


V 


1360 


b{h-x/s)  100(13.5-4.39/3) 


=  1.13  kg/cm2, 


which  is  thus  less  than  the  permissible  value  of  4.5  kg/cm^. 

The  adhesive  stress  of  the  above  reinforcement  at  the  supports  is 


looX  1. 13 

Circumference  of  the  reinforcement    9X1. 0X3. 14 


4.0  kg/cm^ 


(b)  Freely  Supported  Slabs  with  Double  Reinforcement. — The  dimensions 
and  loading  of  the  slab  are  the  same  as  in  the  preceding  example,  so  that 

^  =  78,583. 

Besides  the  lower  reinforcement  consisting  of  9  rods,  10  mm.  diameter,  there 
is  an  upper  layer  of  6  rods  of  10  mm.  diameter.  Then 

F/  =  4,7^,    /i'  =  i.5    (see  Fig.  2,  p.  324). 

The  distance  x  of  the  neutral  layer  below  the  upper  surface  of  the  slab  is  to 
be  computed  from  the  quadratic  equation 

x^  +  2xn^^—  =  2~{hFe  +  h'F/), 


or 


:>;2  +  2X:vXi5^^^^^^-  =  2^(i3.5X7-07  +  i. 5X4.71). 


328  CONCRETE-STEEL  CONSTRUCTION 

Whence 

or 

x  =  4.o^  cm. 

Then  the  compressive  stress  on  the  concrete  is 

6Mx 


Ob 


bx^{T,h  -x)  +  6Fe'n{x-h')  {h-h') 
6X78,583X4.05 


100X4-05^(3  X 13-5 -4-05) +6X4-71  X  15(4-05 -i.5)(i3.5- 1-5) 
The  tensile  stress  in  the  lower  reinforcement  is 


=  26.25  kg/cm-. 


=  12.1  cm., 


^^^^^,(^^^26.25(13.5-4-05)15^    3  ^1^^, 
X  4.05 

and  the  compressive  stress  in  the  upper  reinforcement  is 

^^,^..(^-/0^^  26.25(4.o5-i.5)i5^    g  k 
X  4.05  ^ 

The  distance  between  the  centroids  of  tension  and  compression  in  this  case 
will  be 

M  ^  78,583 

FeOe      7.07  X91I 

whence 

^  ^360  ,    ,  2 

100X12. 1    100X12. 1        ^    ^'  ' 

and  the  adhesive  stress  on  the  lower  reinforcement  at  the  supports  is 

hTQ  100X1.13  1    /  s 

T\  =  —  — —  =  ~  =4.0  kff/cm^ 

Lircumierence  01  remiorcement    9X1. 0X3. 14 


2.  T-Beams 

Simple,  Freely  Supported  T-Beams. — Clear  span  10.60  m.,  distance  between 
mpports  11.00  m.,  live  load  400  kg/m^. 

Load  per  running  meter  of  beam:  Live  load,  400X1.7  =680  kg. 
Asphalt  layer,  30X1.7=  51'' 

Dead  load,  2400(0.25X0.50  +  1.7X0.10)  =708 


Uniform  total  load,  approximately,  1440  kg/m^. 

M  =  ^-  =  i44oX-^Xioo  =  2, 178,000  cm. -kg. 


APPENDIX 


329 


The  reinforcement  consists  of  8  round  rods,  28  mm.  in  diameter  with  F^  = 
49.26  cm-.  The  distance  of  the  neutral  axis  from  the  upper  layer  of  the  slab 
(see  Fig.  4,  p.  325)  is  to  be  computed  by  the  formula: 


that 


and 


or 


2nhFe+hd^ 

^~'2{nFc  +  bdy 

^^2Xi5X54X49.26  +  i7oXio2_  ^ 
2(i5X49-2()  +  i7oXio) 


d  d^ 


10 

19.84-— +■ 


10^ 


2  '  6{2X-d)  2  6(2X19.84-10) 

y  =  15.40. 


d.  o  lo  1 

aso 

/,.0S 

Then  there  results  finally, 
M 


-  lu  6o  - 
-11.00 


r 


6 -0.25 


Fig.  6a  and  66. 


2,178,000 


(Te  = 


Fe{h-x-^y)    49.26(54.0-19.84  +  15.40) 


=  892  kg/cm^, 


(76  = 


OeX  892X19.84 


=  34.5  kg/cm2. 


n{]i-x)  15(54-19.84) 
At  the  supports  the  shear  is  greatest,  being 


F  =  i44oX  =  7632  kg., 

2 


so  that  the  shearing  stress  in  the  concrete  is 
V  7632 


hQ{h-x  +  y)     25(54-19-84  + 15-4) 


6.2  kg/cm^, 


and  the  adhesive  stress  at  the  supports,  on  the  four  round  rods  of  28  mm.  diameter, 
which  extend  to  them,  is 

25X6.2  ,    ,  ^ 

^i=-v7-^ — ^^  =  4-4  kg/cm-. 
4X3.14X2.8 

The  shearing  stress  reaches  the  maximum  permissible  value  of  4.5  kg/cm^ 
at  the  point  at  which 

7632X4-5 


6.2 


=  5540  kg., 


330 


CONCRETE-STEEL  CONSTRUCTION 


that  is,  at  a  point 


^^1^1 — 5540 .^^    ^^^^  p.  ^ 
1440 


and  the  total  diagonal  tension  Zi,  which  must  be  carried  by  the  bent  rods  will  be 

Zi  =  ^(6.2-4.5)XiX25  =  2i8o  kg. 
V  2 


^1  


Fig.  6c. 


Fig.  7. 


When,  therefore,  the  four  upper  rods  of  28  mm.  diameter  are  bent  upward 
through  a  distance  of  1.45  m.,  they  will  carry  a  stress  of  only 


i7e  =  — TTT  7  =  09  Kg/cm'^.* 


4X6.16 


Continuous  T-Beams 


Each  running  meter  of  the  beam  (Fig.  7)  has  to  carry  a  dead  load  of  ^  =  2000 
kg.,  and  a  live  load  of  p  =  1,600  kg.    The  following  moments  therefore  result: 

(a)  At  0.4/  of  the  first  span: 

Mg=  +0.080X2000X6. 752X 100=  +    728,960  cm. -kg. 
—Mp= —0.020X3600X6. 752X 100-- —   328,032  " 
-\-Mp=  +0.100X3600X6.75^X100=  +1,640,160 


so  that 


+  2,369,120 


(b)  Over  the  center  supports: 

Mg=  — 0.10000X2000X6. 752X 100=  —   911,200  cm. -kg. 
—Mp=  —0.11667X3600X6.752x100=  —  1,913,575  " 
+Mp= +0.01667X3600X6.75^X100=  +  273,415 
so  that  Mmax= -2,824,775  " 

(c)  In  the  center  span: 

Mg=  +0.025X2000X6.75^X100=  +    227,800  cm.-kg. 
—lfp=  — 0.050X3600X6.752X100=  —  820,080 
+Mp=  +0.075X3600X6.752X100=  +  1,230,120 

i  +Mmax= +1,457.920 
(  -Mn,ax=-  592,280 


SO  that 


*The  assumption  of  4.5  kg/cm^  for  the  shearing  stress  in  connection  with  the  computation 
of  the  bent  rods  is  not  free  from  objection.  See  in  this  connection,  p.  183  and  187. — (The 
Author). 


APPENDIX 


331 


These  moments  give  the  following  stresses: 
(a)  At  0.4/  of  the  first  span: 

Since  the  girders  are  4.5  m.  apart,  the  permissible  width  of  slab  is 

^=^/3=— J^  =  2.25  m. 
Fe  =  four  round  rods  of  32  mm.  diameter  =32.17  cm-. 

//  =  77  cm.,  ^/=i2  cm.,  /)  =  225  cm.  (see  Fig.  8);  and  the  distance  x  of  the 
neutral  axis  from  the  top  of  the  slab  computed  by  the  formula 

2nhFe  +  bd''^ 


IS 


Further 


y=x 


^     2{nFe  +  bd)' 

2X15X77X32.17  +  225X12^ 
2(15X32.17  +  225X12) 

d  d-^ 


16.8  cm. 


2  6{2x—dy 


and  fmally 


^  ^     12  12^ 

y  =  i6.8  H  

^  2  6(2X16.8-12) 


II. 9  cm., 


M 


2,369,120 


FeQi—x+y)    32.17(77  —  16.8  +  11.9) 


=  1020  kg/cm^, 


oex         1020X16.8  ,    ,  ^ 

<76=— ^  r=     /         A  qT="^9-o  kg/cm^. 

n{h-x)  15(77-16.8) 


r 

' — 

s 

OJO 

k-O.JJ 

 i 

-  -  f 

I- 

Fig. 

8. 

ffl 

Fig.  9 


^,i^z;  N 

\  1j)Q 
.  -  ♦  * 


The  stress  in  the  reinforcement  can  easily  be  reduced  below  1000  kg/cm^,  by 
replacing  one  32  mm.  rod  by  one  of  34  mm.  diameter. 

{b)  Over  an  Intermediate  Support. — Since  the  tensile  strength  of  the  concrete 
is  ignored,  the  floor  slab  receives  no  consideration  in  connection  with  the  nega- 
tive pier  moment,  so  that  in  the  computations  only  a  rectangular  section  (see 
Fig.  9)  of  breadth  ^  =  35  cm.  is  employed. 

_     4X3-2^X-  ,  2X3-4-X-  2 

Fe=^  \  +  =  50-33  ^m2; 

4  4 


^  =  35  cm.,    /^  =  io7cm.,    n  =  \^. 
nFe\        ,    A    ,  2hh\ 


332 

so  that 


CONCRETE-STEEL  CONSTRUCTION 


x  = 


15X50.33 


Further 


35 

:x:  =  49.5  cm. 

2M 


L       ^     15x50.33  J 


Ge 


   2X2,824,775 

^  35X49-5  107 


3 


F.(//-x/3) 


2,824,775 


50.33  107 


49-5 


621  kg/cm^. 


(c)  At  the  middle  of  the  center  span: 

+  i/max= +1,457^920  Cm.-kg. 

3X3.22X7: 

re  =  =24.13  cm^,    6  =  225  cm.,    /^  =  77  cm.,  ^/=i2cm. 


so  that,  as  under  (a). 


2  X  15X24.13X77 +  225X122 

:x;  =  r  — —  =  14-4  cm., 

2(15X24.13  +  225X12) 


12 

:y  =  i4.4  + 


12^ 


M 


Oe- 


Gh- 


2      6(2X14-4  —  12) 
1,457.920 


FeQi-x+y)    24.13(77-14.4  +  9.8) 

GeX  833X14-4  01/  < 

—  -  = —  -  =  i2.8  kg/cm^ 

n{h-x)  15(77-14.4) 

-Mmax=  -592,280  Cm.-kg. 

iX3-4^X7r 


9.8  cm. 

833  kg/cm2; 


=  77  cm.,  F. 
nF 


9.08  cm2,    6  =  35  cm. 


or 


x  = 


15X9-08 


.N- 


2x35x77] 
15x9-08  J' 


ajj 


35  L 
:x:  =  20.9  cm.; 

2M                2X502,280  ,    ,  9 
 r-  =  23.2  kg/cm^ 


Fig.  10. 


Gh  = 


hxih—xj-x)                 I  20.9 
^        '^^  35X20.9(77  


M 


Ge  = 


Fe{h-Xls) 


592,280  ,      ,  „ 

-7-^  r  =  932  kg/cm2. 


9.08  77 


20.9 


The  computation  of  the  shearing  stresses  follows,  as  in  example  2. 


APPENDIX 


333 


B.  COMPRESSION,  COLUMNS 

The  intermediate  supports  of  example  3  (ignoring  the  continuity)  have  to 
carry  a  load  of 

P  =  6. 75(2000  +  3600)  =37,800  kg. 

The  section  is  35X35  cm.,  and  4  round  rods  of  24  mm.  diameter,  with  F^  = 
1 8. 1  cm-,  are  employed.  Then 

7^6  =  1225  cm^. 
Fe=i8. 1  cm2. 

P  37,800  ,    ,  „ 

^^  =  -E^-T~^  =  7-        o     =25.3  kg/cm2, 

Fb  +  nFe    1225  +  15X18.10     ^  ^   &/  ' 

(76=^(76  =  25.3X15=380  kg/cm-. 


REGULATIONS  OF  THE  ROYAL  PRUSSIAN  MINISTRY  OF 
PUBLIC  WORKS,  FOR  THE  CONSTRUCTION  OF  REIN- 
FORCED  CONCRETE  BUILDINGS.    MAY  24,  1907 

L  GENERAL 

A.  TESTING 
Sec.  1 

1.  The  construction  of  buildings  or  their  structural  parts  in  reinforced  con- 
crete is  to  be  subject  to  special  supervision  by  the  building  inspectors.  For  this 
reason,  when  application  is  made  for  a  permit  for  a  structure  in  whole  or  in 
part  of  reinforced  concrete,  drawings,  statical  calculations  and  specifications 
must  be  submitted  in  which  the  general  plans  and  all  important  details  are  shown. 

In  case  the  owner  or  contractor  does  not  decide  upon  the  use  of  reinforced 
concrete  until  the  work  is  under  way,  the  building  inspectors  must  insist  that 
the  above  described  drawings  for  the  reinforced  concrete  work  be  filed  a  sufficient 
period  before  commencing  the  construction.  Under  no  circumstances  is  work 
to  be  begun  before  permission  therefor  has  been  granted. 

2.  The  specifications  must  state  the  source  and  the  kind  of  concrete  aggregates 
to  be  used,  their  proportions,  the  amount  of  water,  and  the  compressive  strength 
which  is  to  be  developed  by  30  cm.  (11.8  in.)  cubes,  28  days  old,  made  at  the 
building  site,  and  of  the  materials  stated.  If  required  by  the  building  inspectors, 
the  compressive  strength  must  be  shown  by  test  before  commencing  work. 

3.  The  concrete  must  be  mixed  in  proportions  by  weight;  a  bag  of  57  kg. 
(125.4  lbs.),  or  a  barrel  of  170  kg.  (374  lbs.)  of  cement  being  the  unit  of  measure. 


334 


CONCRETE-STEEL  CONSTRUCTION 


The  aggregates  may  be  either  weighed,  or  measured  in  vessels  the  capacity  of 
which  has  previously  been  so  arranged  that  the  weight  corresponds  with  the 
proportions  already  determined. 

4.  The  contract  is  to  be  signed  by  the  owner,  and  by  the  general  contractor 
and  the  special  contractor  who  is  to  do  the  work.  The  building  inspectors  are 
to  be  notified  of  any  change  of  contractors. 

Sec.  2 

1.  The  quality  of  the  concrete  materials  is  to  be  certified  by  an  official  test- 
ing laboratory.    As  a  rule,  such  certificates  must  not  be  more  than  a  year  old. 

2.  Only  such  Portland  cement  may  be  used  as  fulfils  the  Prussian  require- 
ments. The  certificates  of  its  quality  must  give  particulars  as  to  its  constancy 
of  volume,  time  of  set,  fineness,  as  well  as  of  its  tensile  and  compressive  strength. 
The  constancy  of  volume  and  time  of  set  must  be  independently  tested  by  the 
builder. 

3.  Sand,  gravel  and  other  aggregates  must  be  proper  for  the  manufacture  of 
concrete,  and  the  special  purpose  intended.  The  size  of  the  particles  must  be 
such  that  the  placing  of  the  concrete,  and  its  tamping  between  the  reinforcing 
rods  and  between  them  and  the  forms  can  be  done  with  certainty  and  without 
displacing  the  steel. 

Sec.  3 

1.  The  method  of  computation  must  provide  at  least  as  much  security  as 
that  provided  according  to  the  Leitsatze,"  section  II,  and  according  to  the  methods 
of  calculation  with  examples  in  section  III  of  these  Regulations.  This  must  be 
demonstrated  by  the  contractor,  if  required. 

2.  In  the  case  of  new  types  of  construction,  the  building  inspectors  may  condi- 
tion the  permit  on  the  results  of  preliminary  test  structures  and  loading  exper- 
iments.   The  latter  are  to  be  carried  to  the  point  of  failure. 

B.  CONSTRUCTION 

Sec.  4 

1.  The  building  inspectors  may  have  the  quahty  of  the  materials  in  use  in  the 
work  tested  by  an  official  laboratory,  or  in  any  other  manner  deemed  suitable 
by  them,  and  can  also  have  tested  the  strength  of  the  concrete  made  therefrom. 
The  strength  tests  can  also  be  made  at  the  building  site  by  a  concrete  press,  the 
accuracy  of  which  has  been  certified  by  an  official  testing  laboratory. 

2.  The  specimens  to  be  so  tested  must  be  cubes,  30  cm.  (11.8  in.)  on  an  edge. 
These  specimens  are  to  be  dated,  sealed  for  identification,  and  stored  until  they 
have  properly  hardened,  in  accordance  with  the  instructions  of  the  building 
officials. 

3.  The  cement  must  be  delivered  in  the  original  package  at  the  point  of 
consumption. 


APPENDIX 


335 


4.  The  concrete  must  be  mixed  in  such  manner  that  the  quantities  of  the 
several  constituents  always  agree  with  the  specified  proportions,  and  can  always 
be  readily  determined.  Where  measuring  vessels  are  employed,  they  are  always 
to  be  filled  in  the  same  manner  and  to  the  some  degree  of  compactness. 

Sec.  5 

1.  As  a  rule  the  manipulation  of  the  concrete  must  commence  immediately 
after  it  is  mixed,  and  must  be  completed  before  it  has  begun  to  set. 

2.  In  warm,  dry  weather,  the  concrete  must  not  lie  unused  longer  than  one 
hour,  and  in  cooler  damp  weather,  longer  than  two  hours.  Concrete  that  is  not 
immediately  placed,  must  be  protected  from  climatic  influences,  such  as  sun, 
wind  and  heavy  rain,  and  must  be  re-mixed  before  use. 

3.  The  manipulation  of  the  concrete  must  always  be  continuous  until  the  tamp- 
ing is  complete. 

4.  The  concrete  must  be  laid  in  layers  not  more  than  15  cm.  (6  in.  approx.) 
thick,  and  should  be  tamped  as  much  as  the  amount  of  water  present  will  permit. 
Rammers  of  suitable  form  and  weight  are  to  be  used  for  the  tamping. 

Sec.  6 

1.  Before  placing,  the  reinforcement  is  to  be  cleaned  of  all  loose  rust,  dirt 
and  grease.  Special  care  is  to  be  exercised  to  see  that  the  reinforcement  is  in  proper 
position,  that  the  rods  are  of  correct  form  and  are  properly  spaced  and  kept  in 
place  by  necessary  contrivances,  and  are  completely  enveloped  in  special  fine  con- 
crete. Where  the  reinforcement  in  the  beams  is  in  several  layers,  each  one  must 
be  separately  embedded.  Beneath  the  reinforcement  must  be  a  layer  of  concrete 
at  least  2  cm.  (J  in.  approx.)  thick  in  beams,  and  at  least  i  cm.  (|  in.  approx.) 
thick  for  slabs. 

2.  The  forms  and  supports  of  the  floors  and  beams  must  be  strong  enough 
to  resist  bending,  and  be  solid  enough  to  withstand  the  tamping.  The  forms 
are  to  be  so  devised  that  they  can  be  removed  without  disturbing  such  supports 
as  may  be  necessary  until  the  concrete  has  properly  set.  As  far  as  possible,  only 
unspliced  lumber  is  to  be  used  for  supports.  If  splices  are  unavoidable,  the 
supports  must  be  strongly  and  firmly  connected  at  the  joints. 

3.  Column  forms  must  be  so  constructed  that  the  depositing  and  compacting 
of  the  concrete  can  be  done  through  one  open  side,  which  is  closed  as  the  work 
progresses,  so  that  it  may  be  closely  inspected. 

4.  At  least  three  days'  notice  must  be  given  the  building  inspectors  l^efore 
the  completion  of  the  forms  and  the  proposed  commencement  of  the  concreting 
for  each  story. 

Sec.  7 

1.  As  far  as  possible,  the  several  concrete  layers  must  be  deposited  on  fresh 
material;  but  in  every  case,  the  top  of  the  old  layer  must  be  roughened. 

2.  When  w^ork  is  to  be  done  on  hardened  concrete,  the  old  toj)  surface  must 
first  be  roughened,  swept  clean,  wetted,  and  coated  with  a  thin  cement  grout  before 
new  material  is  deposited. 


336 


CONCRETE-STEEL  CONSTRUCTION 


Sec.  8 

In  the  construction  of  walls  and  columns  in  buildings  several  stories  in  height, 
work  on  th^  upper  members  can  be  continued  only  after  the  structural  parts  in  the 
lower  stories  have  become  sufficiently  hard. 

Sec.  9 

1.  During  freezing  weather,  work  can  be  carried  on  only  under  such  condi- 
tions as  will  preclude  the  possibility  of  injury  by  the  frost.  Frozen  materials  must 
not  be  used. 

2.  On  the  advent  of  mild  weather,  after  a  prolonged  freeze  (Sec.  ii),  work 
can  be  continued  only  after  permission  so  to  do  has  been  obtained  from  the 
building  inspectors. 

Sec.  10 

1.  Until  the  concrete  has  properly  hardened,  the  structural  parts  must  be 
protected  against  the  effects  of  frost  and  against  premature  drying  out,  as  well 
as  against  vibration  and  loading. 

2.  The  interval  which  must  elapse  between  the  completion  of  the  tamping, 
and  the  removal  of  the  forms  and  supports  must  depend  upon  the  prevailing 
weather,  the  spacing  of  supports  and  the  weight  of  the  members.  The  side  forms 
of  beams,  column  forms,  and  floor  slab  forms  must  not  be  removed  in  less  than 
eight  days,  and  the  supports  of  the  beams  in  not  less  than  three  weeks.  With 
large  spacings  of  columns  and  areas  of  members,  the  interval  must  be  extended 
to  six  weeks. 

3.  In  many-storied  buildings,  the  supports  under  the  lower  slabs  and  beams 
can  be  removed  only  when  the  hardening  of  the  upper  ones  has  so  far  advanced 
that  they  can  support  themselves. 

4.  When  the  tamping  was  completed  but  a  short  time  before  frost  occurred, 
the  removal  of  forms  and  supports  is  to  be  done  with  the  greatest  care. 

5.  Should  freezing  take  place  during  the  period  of  hardening,  in  view  of  the 
fact  that  the  latter  may  have  been  retarded  by  the  frost,  the  intervals  mentioned 
in  section  2  should  be  extended  by  at  least  the  frozen  period. 

6.  For  the  removal  of  forms  and  supports,  special  contrivances  (wedges,  sand 
boxes,  etc.)  must  be  employed  to  prevent  shock. 

7.  The  building  inspectors  must  be  given  at  least  three  days'  notice  of  the 
removal  of  forms  and  supports. 

Sec.  11 

A  diary  must  be  kept  of  the  progress  of  the  work,  and  it  must  always  be 
open  for  inspection  at  the  building  site.  Freezing  weather  is  to  be  specially  noted 
therein,  together  with  the  temperatures  and  the  hour  of  their  observation. 


APPENDIX 


337 


C.  REMOVAL  OF  FORMS 

Sec,  12 

1.  The  structural  parts  must  be  exposed  at  different  points  as  required  by 
the  building  inspectors,  so  that  the  character  of  the  work  may  be  determined. 
The  right  is  reserved  to  determine  by  tests  the  hardness  and  strength  of  ques- 
tionable parts. 

2.  Should  question  arise  as  to  the  proportions  and  degree  of  hardness  of  any 
portion,  test  samples  may  be  taken  from  the  finished  parts. 

3.  Where  load  tests  are  considered  necessary,  they  are  to  be  conducted  accord- 
ing to  the  directions  of  the  proper  officials.  The  owner  and  the  contractor  will 
be  given  due  notice,  and  those  interested  informed.  Loading  tests  should  be  made 
not  less  than  45  days  after  the  concrete  has  set,  and  be  restricted  to  the  portion 
deemed  necessary  by  the  building  inspectors. 

4.  In  load  tests  of  slabs  and  beams,  the  following  method  is  to  be  adopted: 
In  loading  a  whole  floor  panel,  if  g  is  the  dead  load,  and  p  the  uniformly  dis- 
tributed live  load,  the  applied  load  is  not  to  exceed  0.5  ^  +  1.5  p.  With  live  loads 
of  more  than  1000  kg/m^  (205  lbs/ft^),  simply  the  live  load  may  be  used.  If  only 
a  strip  of  the  floor  is  to  be  tested,  the  load  is  to  be  uniformly  distributed  over  a 
space  at  the  center  of  the  slab,  the  length  of  which  is  equal  to  the  span,  and  a  third 
of  the  span  in  breadth,  but  never  less  than  i  meter  (3.3  ft.).  The  applied  load 
in  this  case  is  not  to  exceed  a  value  of  ^  +  2  p.  The  dead  load  is  that  of  all 
the  structural  parts,  including  the  weight  of  the  flooring  and  slab,  the  live  load 
being  of  the  character  specified  in  Sec.  16,  No.  3. 

5.  In  any  loading  tests  of  columns,  unequal  settlement  of  the  structural 
parts  and  an  overloading  of  the  subsoil  is  to  be  avoided. 

II.  RECOMMENDATIONS  FOR  STATICAL  COMPUTATIONS 

A.  DEAD  LOAD 
Sec.  13 

1.  The  weight  of  the  concrete,  including  the  reinforcement  is  to  be  assumed 
at  2400  kg/m3  (149  lbs/ft^),  unless  otherwise  specifically  stated. 

2.  With  that  of  the  slab  is  also  to  be  included  the  weight  of  the  material 
forming  the  finish,  determined  according  to  known  units. 

B.  DETERMINATION  OF  EXTERNAL  FORCES 
Sec.  14 

I.  In  members  subjected  to  flexure,  the  applied  moments  and  reactions  are 
to  be  computed  according  to  the  usual  rules  for  the  kind  of  loading  and  manner 
of  support  of  freely  supported  or  continuous  beams. 


338 


CONCRETE-STEEL  CONSTRUCTION 


2.  In  freely  supported  slabs,  the  clear  span,  plus  the  thickness  of  the  slab  at 
the  center,  is  to  be  considered  as  the  length  in  computations,  and  in  continuous 
slabs  the  distance  between  centers  of  supports  is  to  be  employed.  In  beams, 
the  length  is  to  be  considered  as  the  free  span  increased  by  the  width  of  the 
supports. 

3.  In  slabs  end  beams,  continuous  over  several  spans,  where  the  moments 
and  reactions  ccnnot  be  computed  according  to  the  rules  for  continuous  members, 
under  the  assumptions  of  free  supports  at  ends  and  intermediate  points,  or  cannot 
be  determined  by  experiment,  the  bending  moments  at  the  centers  of  spans  are 
to  be  taken  as  four-fifths  of  the  value  for  a  slab  resting  freely  on  two  supports. 
Over  the  supports,  the  negative  moments  are  to  be  the  same  as  the  span  moments 
when  both  sides  are  freely  supported.  Beams  and  slabs  are  to  be  computed  as 
continuous  according  to  these  rules  only  when  the  supports  are  rigid  and  at  the 
same  level,  or  consist  of  reinforced  concrete  beams.  In  arranging  the  reinforce- 
ment, under  all  circumstances  the  possibility  of  the  occurrence  of  negative  moments 
is  to  be  considered. 

4.  In  beams,  moments  from  end  restraint  are  to  be  included  in  computations 
only  when  special  structural  conditions  make  true  restraint  possible. 

5.  Continuity  is  not  to  be  assumed  over  more  than  three  spans.  With  live 
loads  of  more  than  1000  kg/m^  (205  lbs/ft^)  the  calculations  are  to  be  made  for 
the  most  unfavorable  arrangement  of  load. 

6.  In  computations  of  T-beams,  the  width  of  each  flange  at  the  center  of  span 
is  not  to  be  assumed  as  more  than  one-sixth  the  length  of  the  beam. 

7.  Rectangular  slabs,  freely  supported  on  all  sides,  with  double  reinforcement 

and  uniformly  distributed  load  can  be  computed  according  to  the  formula,  M  =  - — , 

when  the  long  side  a  is  less  than  one  and  a  half  times  the  breadth  h.  Special 
forms  and  distribution  of  reinforcing  rods  are  to  be  employed  to  care  for  negative 
moments  at  supports. 

8.  Even  when  so  computed,  thickness  of  slabs  and  of  the  flanges  of  T-beams 
is  never  to  be  less  than  8  cm.  (3.1  in.). 

9.  In  columns,  the  possibility  of  eccentric  loading  is  to  be  considered. 

C.  DETERMINATION  OF  INTERNAL  FORCES 
Sec.  15 

1.  The  modulus  of  elasticity  of  the  reinforcement  is  to  be  assumed  as  fifteen 
times  that  of  the  concrete,  unless  otherwise  specified. 

2.  The  stresses  in  any  section  of  a  body  under  flexure  are  to  be  computed 
on  the  assumption  that  the  strains  are  proportional  to  the  distances  from  the 
neutral  axis,  and  that  the  reinforcement  carries  all  the  tension. 

3.  In  buildings  or  members  exposed  to  the  weather,  to  dampness,  to  smoke 
gases,  and  similar  deleterious  influences,  it  must  be  shown  that  cracks  wiU  not 
occur  from  the  tensile  stress  to  which  the  concrete  is  subjected. 

4.  Shearing  stresses  are  to  be  considered,  unless  the  form  and  arrangement 
of  the  members  show  their  harmless  nature.    They  must  be  carried  by  a  proper 


APPENDIX 


339 


arrangement  of  the  reinforcement,  whenever  the  design  of  the  structure  is  not 
made  to  care  for  them. 

5.  As  far  as  possible,  the  reinforcement  must  be  so  constructed  as  to  preclude 
its  displacement  in  the  concrete.  Adhesive  stresses  should  always  be  susceptible 
of  mathematical  determination. 

6.  Columns  should  always  be  computed  with  regard  to  buckling  unless  the 
length  is  less  than  eighteen  times  the  least  lateral  dimension.  The  spacing  of 
the  longitudinal  reinforcement  is  to  be  maintained  by  cross  ties.  The  spacing 
of  the  ties  must  not  exceed  the  least  dimension  of  the  column  or  be  more  than 
thirty  times  the  diameter  of  the  reinforcing  rods. 

7.  In  computing  columns  with  regard  to  buckling,  the  Euler  formula  is  to  be 
employed. 

D.  PERMISSIBLE  STRESSES 
Sec.  16 

1.  In  members  subjected  to  flexure,  the  compressive  stress  of  the  concrete 
should  be  one-sixth  of  its  ultimate  value,  and  the  compressive  and  tensile  stresses 
in  the  reinforcement  should  not  exceed  1000  kg/cm^  (14,223  lbs/in^). 

2.  If  the  tensile  stress  in  the  concrete  must  be  considered  as  required  in  Sec. 
15,  No.  3,  two-thirds  of  the  tensile  strength  of  the  concrete  as  determined  by 
experiment,  may  be  allowed.  When  tension  tests  are  wanting,  the  tensile  stress 
is  not  to  be  assumed  greater  than  one-tenth  the  ultimate  compressive  strength. 

3.  The  following  loading  values  are  to  be  assumed: 

{a)  In  members  subject  to  moderate  vibration,  as  in  the  floors  of  dweflings, 
stores,  warehouses,- -the  actual  dead  and  live  loads. 

(h)  In  members  subject  to  more  violent  vibration  or  to  widely  fluctuating  loads, 
as  in  the  floors  of  places  of  assembly,  dance  hafls,  factories,  storehouses — the  dead 
load — plus  the  live  load  increased  50%. 

{c)  In  loads  attended  by  violent  impact,  as  in  the  roofs  of  ceflars  under  drive- 
ways and  courtyards — the  dead  load  plus  the  live  load  increased  100%. 

4.  In  columns,  the  concrete  should  not  be  stressed  to  more  than  one-tenth  its 
ultimate  strength.  In  computing  the  reinforcement  with  regard  to  buckling,  a 
factor  of  safety  of  five  is  to  be  employed. 

5.  The  shearing  stress  in  the  concrete  is  not  to  exceed  4.5  kg/cm^  (64  lbs/in^.). 
If  a  greater  shearing  strength  is  possible,  the  permissible  stress  is  not  to  be  more 
than  one-fifth  of  the  ultimate  strength. 

6.  The  adhesive  stress  is  not  to  exceed  the  permissible  shearing  stress. 


340 


CONCRETE-STEEL  CONSTRUCTION 


III.  METHODS  OF  CALCULATION,  WITH  EXAMPLES 

A.  SIMPLE  BENDING 
(a)  Without  reference  to  the  tensile  stress  in  the  concrete. 

With  single  reinforcement  of  total  area  /,  of  a  beam  or  slab  of  width  b,  if  the 
ratio  of  the  moduli  of  elasticity  of  the  steel  and  the  concrete  is  represented  by 
n,  then  the  distance  of  the  neutral  axis  below  the  top  is  given  by  the  equation  of 
the  statical  moments  of  the  elemental  areas  about  this  axis  (see  Fig.  i). 


~^=nfe{h—a—x)  (i) 


is 


nfe 


(2) 


From  the  equality  of  the  moments  of  the  external  and  internal  forces,  it  will 
follow  that 


M  =  oi,^h(h  —  a—^^=  ocfc  (^h  —  z  —   ^ : 


(3) 


wherein  indicates  the  maximum  concrete  compressive  stress,  and  the  average 
steel  tension.    From  this  there  follows 


2M 


Ob 


bxi  h  —  a 


M 


Fig.  I. 


(4> 


(5)' 


Under  certain  circumstances  the  following  easily  obtained  equations  are  of 
value 

n{h—a)ai 


bx 

a}j  =OeJe 

2 


(6> 
(7) 


In  T-shaped  sections,  such  as  T-beams,  the  calculation  does  not  differ  from 
that  above  when  the  neutral  axis  falls  within  the  slab  or  at  its  lower  edge. 

If  the  neutral  axis  is  in  the  stem,  the  small  compression  in  the  stem  may  be 
ignored. 


APPENDIX 


341 


Then  (see  Fig.  2) 


x-d 

Ou  =  Oo^\ 


h  —a—x 
(Jc  =  n  -Oo', 


(To +(7: 


bd  =  Ocje, 


(8) 

(9) 
(10) 


or,  by  introducing  into  equation  (10)  the  values  of  cr,^  and  from  equations 
(8)  and  (9) 

 \-nfe{h  —  a) 


x=- 


bd+nfe 


(II) 


r — 4 


' — Be- 


I 

Fig.  2. 


Since  the  distance  of  the  centroid  of  the  compression  trapezoid  below  the 
top  is 

d  Oo-\-2au 


x—y-= 


3   Oo^Ou  ' 

there  is  obtained  by  introducing  the  value  of     from  ecpation  (8) 

d         d?        2  (  (x-dy^' 

y=x  Htt  -K  =  -[^  +  -7 

2     o{2x—d)    3  \  2x—a 


M 


Ge- 


.     .     .  (12) 

.     .     .  (13) 

.     .     .  (14) 

n{h-a-x) ^^^^ 

When  beams  and  slabs  also  contain  top  reinforcement,  the  following  equa- 
tions may  be  employed: 

For  the  location  of  the  neutral  axis: 

bx'^ 

—  —fe{x—a)-\-nfe{x—a)=nfe{h—a—x),   (16) 


fe{h  —  a—x-\-y)' 

XOo 


from  which 


=   ^  -  +   -j,  -j  +-^l{n-i)f/a  +  nMh-a)\.   .  ( 

*  <Tj^  =  concrete  stress  at  under  side  of  slab. — Trans. 
<7o  =  concrete  stress  at  top  of  slab. — Trans. 


17) 


342 


CONCRETE-STEEL  CONSTRUCTION 


For  the  applied  moment 

boc    /  X\ 
M=—ab\h—a—-J—fe'ab'{h  —  2a)+fe'cie'{h  —  2a). 


(i8) 


Here  oh'  represents  the  concrete  compressive  stress  at  the  average  height  of 
the  top  reinforcement,  and  is  determined  by 


Since,  further, 


,    x  —  a 

Gb  =  Ob. 


,    nix— a) 

Oe  —  Gb, 


it  will  follow  that 


\hx  ( ,  x\  ,  ^,,x—a,,  1 
M=Y~i^i-a--J^{n-i)fe'-—{h-2a)^ 


Ob. 


(19) 


If  the  slight  reduction  in  the  area  of  the  compressed  concrete  made  by  the 
reinforcement  is  ignored,  equation  (17)  becomes 


x=  - 

and  (19)  becomes 


n{fe+f. 


M 


Oh* 


(20) 


(21) 


FlG.  3. 


Fig.  4. 


Fig.  5. 


If  (7^  has  been  computed  by  equation  (21)  for  a  given  moment  the  stresses 
and  oj  are  easily  determined  from  the  law  that  the  stress  is  proportioned  to 
the  distance  from  the  neutral  axis.    If  the  maximum  moment  has  been  deter- 
mined for  a  permissible  concrete  compressive  stress  cr^,  the  stresses  o^  and  oJ  are 
to  be  found  from 


M 


=feGe(  h  —  a  

\  3 


±  Je'Oe'\   a 


22) 


or  smce 


,  x—a 

Oe  =-j  Oe. 

h—a—x 


APPENDIX 


343 


The  common  centroid  of  the  concrete  and  the  reinforcement  in  the  zone  of 
compression  may  be  determined  from 


yi 


bx  2  hx^ 

 XGb  +  (Tc'U(x—a)  \-nfe'(x  —  a)^ 


hx 


hx^ 


(24) 


+  n//(x  —  a) 


and  then 


M=/e(7e{h—a—x-\-yi)  (25) 


b.  Taking  into  consideration  the  concrete  tensile  stresses. 

With  single  reinforcement,  the  equation  corresponding  to  equation  (1),  (sec 
Fig.  5),  is 

bx^_b{h-x)'^ 
2 

so  that 


+nfe  (h—a—x),  (26) 


—+nfe{h—a) 


(27) 


bh  +nfe 

From  the  equality  of  the  tensile  and  compressive  forces,  there  follows 

bx        .h—x  .  . 

 Ohd  =  0^—ahz-[-Oefc,  (28) 

2  2 


and  from  the  proportionality  of  stress  and  strain 


h—x 

Ghz  =  Obd, 


Ji  —a—x 

Ge^n  Obd. 


(29) 

(29a) 


The  moment  equation  for  the  neutral  axis  is 


bx      2         //  —  X  2 
M = —Obd  —x-^b  obx  - (Ji  —x)+  Oefeih  —  a—x), 

23  23 


from  which  there  follows,  with  the  help  of  equations  (29)  and  (29^)5 


Obd 


bx^  b(h-x)^ 


+nfc(h  —a—x)' 


(30) 


(31) 


When  M  is  given,  <7^^  can  be  determined  from  equation  (31),  and  then  with 
equations  (29)  and  (29^),       and  o^. 

In  T-beams,  when  the  neutral  axis  passss  through  the  stem: 


//2  d~ 

bi-  +  (b-bi)~-  +  nMh-a) 
2  2 


x=- 


bih  +  {b-bi)d  +  nfe 


(32) 


M=b^^^^^dy+bi—  —{x—d)^-\-bi- — -obz  —{h—x)-\-Oefe{h—a—x),  (33) 
2  23  23 


344 


CONCRETE-STEEL  CONSTRUCTION 


M 


=-^^^d{2X-d)y  +  ^j^^^  .  (33a) 

 (34) 


h  —X 


h  —  a—x 

Oe  =  fl  Oq. 


(34^J) 


These  equations  are  very  inconvenient  for  the  determination  of  the  dimensions 
of  sections  for  a  given  appHed  moment.  If  h,  hi,  h  and/g  are  given,  and  the  assump- 
tion is  made  that  the  neutral  axis  coincides  with  the  lower  edge  of  the  slab,  then 


hx''    ,  {h-xY-      ...  . 

—  =0i  \-nfe(h—a—x), 

2  2 


whence 


^^^x^  +  (61  /z  +  nfe)  X  =  ~-  +  nfe  ill — a) . 


(35) 


(36) 


From  this  x  is  found  and  the  thickness  of  slab.  The  stresses  which  occur  are 
found  from 


bx^  {Ji—xY 

 \-0i  h 

.  3  3 


(37) 


and  from  equations  (34)  and  (34^1). 

When  reinforcement  is  introduced  also  into  the 
zone  of  compression,  there  are  found  for  both 
beams  and  slabs  (see  Fig.  6) 


+  {n-i)[fe'a+fe{h-a)] 
bh  +  {n-i){f/  +  fe) 


bj^^HhjxY  ^    _  _^)2  4.y^(/,  _^  _^)2) 

3  3 


(Tbd 
X  ' 


If  the  upper  and  lower  reinforcement  are  of  equal  area,  x-^—  and 


M- 


bh^  ,  4(w-i)/.//z 


(38) 
(39) 

(40) 


For  the  usual  types  of  structural  members,  slabs  and  beams  of  rectangular 
section  and  with  reinforcement  only  on  the  tension  side,  simplifications  of  the 
expressions  (2)  (4)  and  (5)  may  be  made.  If  the  apphed  moment  as  well  as 
the  sections  of  the  concrete  and  of  the  reinforcement  are  given,  and  if  it  is  desired 

to  find  the  resulting  stresses,  the  relation /e  =  ^^——  may  be  assumed,  wherein 

w  =  — — —  may  be  found  for  different  conditions.    For  various  values  of  w, 

Table  A  (Appendix)  may  be  employed  for  the  determination  of  the  proper  values 
of  X,  Oh  and  (7e. 


APPENDIX 
Table  A 


345 


Value  o 
fc 


Corresponding 
Value  of  X. 


Stress 
Ob 


Stress 
Or 


b(h  —  a 


lOO 

hih  —  a 


I  lO 

h{h—a 


1 20 
h(h-a 


b{h  —  a 


140 

h{h-a 


150 
b{h-a 


160 
b(h-a 


b(h-a 


180 
b(h-a 


lyo 
b{h-a 


o.4i8(A  — (/ 
0.403 (A — a 
0.391  (h  —  a 
o.379(A— (/ 
o.368(^-a 

o.349(/z  — a 
o.34i(/z  — cz 

o.^26(h  —  a 
0.320(7^ — a 


5-559- 


5-735' 


5. 895' 


6.040' 


6.194. 


6-344- 


6.485- 


6.617 


6.756. 


b{h-a)- 

M 
b{Ji-a)- 

M 
b{h-af 

M  

bih-af 

~  M 
b{h  —  af 

M 
b{Ji-af 

M 
b'ih^a)- 

M 
b{h-a)'' 

M 


b{h—a) 
M 

6-883-- 


b{h-af 

M 
b{h-ay 


ii6- 


138 


149 


160 


b{h-a)- 
M 

'bih^y" 

M 

b{h-a)''~ 
M 

b{h—a)-~ 
M 

bih^^'" 
M 


'  b{h-a)-' 


181 


192 


203 


224 


b{h-a) 

M 
b{h-af 
M 

b^Ii^^-' 
M 

'b{h—af 
M 

b{h^)-' 


20.867^7/, 
22. 145a/, 
23.409^7;, 
24.668r7^ 
25.83lrT6 

26.797^75 

27.91 1<76 
29.0l6rrft 
30.049^6 
30.946^7/, 
32. 000^75 


If  the  dimensions  are  sought,  when  the  appHed  moments  and  the  concrete 
and  steel  stresses  are  given,  x  is  first  found  from  equation  (6),  x=s  {h—a),  where 
nob 


5  = 


Of  +  nob 


This  value  inserted  in  equation  (4)  gives 


h-a 


(41) 


The  expression  for  />,  from  equation  (5)  is 


Oe 


h  —  a  — 


s{h-a) 


or  when  // 


1/   ,  . 


this  becomes 


-VMb^tVMb.     ,  , 


(42) 


346 


CONCRETE-STEEL  CONSTRUCTION 


The  values  of  x,  h  —  a,  and  Z^,  found  according  to  diis  method  for  different 
values  of      and      may  be  tabulated.* 

Such  a  table  may  also  be  used  for  T-beams,  when  the  neutral  axis  coincides 
with  the  under  side  of  the  slab,  or  when  such  a  location  of  the  axis  is  made  a  condi- 
tion of  the  design. 

B.  CENTRAL  LOADING 

If  F  is  the  area  of  the  concrete  under  pressure,  and  is  the  total  area  of  rein- 
forcement, the  permissible  load  will  be 


so  that 


P  =  {F  +  nfe)ob,  (43) 

nP  ^  ^ 

,^  =  nab=y^^^  (45) 


C.  ECCENTRIC  LOADING 

The  calculations  are  to  be  made  as  for  homogeneous  materials,  the  area  of 
reinforcement  being  replaced  by  an  equivalent  concrete  area  w-times  larger  in  all 
expressions  for  areas  of  sections  and  moments  of  inertia.  Tensile  stresses  which 
may  occur  are  to  be  cared  for  by  reinforcement. 


D.  EXAMPLES  « 

The  maximum  stresses  in  the  steel  and  the  concrete  are  to  be  ascertained 
for  a  freely  supported  dwelling  floor  of  2  m.  span 


 j.      _  cm   J 

^   1;^^^,,^^^.^  2^nd   TO   cm.   thick,   reinforced   with  5.02  cm^/m 

^<  2  00  

^  p        width  (10  rods  of  8  mm.  diameter)  placed  1.5  cm. 

y  from  the  bottom  of  the  slab  to  the  centers  of  the 

rods. 

The  dead  weight  of  the  floor  per  m^  is  0.10X2400   240  kg. 

Over  which  is  placed  10  cm.  of  rolled  cinders   60  kg. 

A  wooden  floor  3.3  cm.  thick  with  stringers   20  kg. 

A  finish  1.2  cm.  thick   20  kg. 

Live  load   250  kg. 

Total  ^   590  kg. 

*  The  original  contains  such  a  table  with  steel  stresses  varying  from  14,223  to  11,379  lbs/in^, 
and  concrete  stresses  from  640  to  284  lbs/in^  This  table  has  not  been  translated  because 
these  values  are  so  far  below  usual  American  practice  that  it  would  be  of  no  practical  value. 
—(Trans.) 


APPENDIX 


347 


Then 


590X2. i-Xioo  , 
M  =  ^  g  =  32,500  kg.-cm.; 


^^15X5-02 
100 


r  /   2x100x8.5  1 

VI  +—  ^  — I    =2.0  cm. 

15x5-02    J  ^ 


2X32,500        _ .  ,  2 

ah=  't-^^  -  =  20.8  kg/cm2; 

100X2.9(8.5-0.97)  ^ 

(Te  =  r  =  86o  kg/cm2. 

5.02(8.5-0.97) 

The  concrete  compressive  stress  of  29.8  kg/cm^  is  permissible  if  the  concrete 
employed  has  an  ultimate  compressive  strength  of  6X29.8  =  178.8  kg/cm^. 
By  using  Table  i,  since  7^=5.02,  there  are  found, 

100X8.5 

m=   =  170  approx. ; 

5.02 

6.617X32,500  01/2 
100X8.52  =29.8kg/cm2; 

(7c  =  29.01 6X29.8  =  865  kg/cm^. 

500  X  2.00 

To  ascertain  the  shear  and  adhesive  stresses  at  the  supports,  V  =  ~ — ^  

=  590  kg.  must  be  found.    Then  the  shearing  stress  is 

ro  =  -r-^  r  =  7^^^  r=o.78  kg/cm^. 


h[li 

The  adhesive  stress  is  then 


in  which  11  represents  the  circumference  of  the  reinforcement. 

100X0.78  -    ,  2 

Ti  =  ;7 —  =  ^.io  kg/cm-^. 

^    10X0.8X3. 14 

Neither  shearing  or  adhesive  stresses  reach  the  maximum  permissible  limits. 

2.  A  simply  supported  T-beam,  with  single  reinforcement  is  assumed,  with 
a  span  of  2  m.  The  live  load  is  1000  kg/m^,  for  a  factory.  The  necessary  size 
of  concrete  and  reinforcement  is  to  be  ascertained  on  the  assumption  that  the 
concrete  employed  will  develop  a  compressive  strength  of  180  kg/cm^. 

For  the  calculation  of  the  dead  load,  the  thickness  of  the  slab  will  be  tenta- 
tively assumed  as  18  cm.,  so  that  the  total  span  considered  is  2  18  m. 

The  dead  weight  of  the  slab  per  sq.m.  is  0.18X2400=  .     . .  432  kg. 

The  covering  of  cinders  20  cm.  thick   120  kg. 

Cement  finish  2.5  cm.  thick   48  kg. 


Total 


600  kg. 


348 


CONCRETE-STEEL  CONSTRUCTION 


^,  6oo  +  i.t;Xiooo  1 

The-n  M  =  ^  X  2.182x1 00  =  124, 700  kg. -cm. 

8 

Since  ^7&  =  -^  =  3o  and  (7e  =  iooo  kg/cm^  are  the  permissible  stresses,  accord- 


ing to  equation  (6), 

15X30 


-{h—a)  =o.^i{h—a), 


1000 -[-15X30 
and  then  by  equation  (41), 


According  to  equation  (i)  fe  is  found  to  be 


o.^iN  '  100 

I  ^5-)o.3iX3o 


24,700 

17.3  cm. 


bx^  100X0.312X  17.32        .  ^ 

je  =  — 77  \^~t: — 7  cm^ 

2n{h-a-x)  2X15(17.3-0.31X17-3) 

Nine  round  rods,  11  mm.  in  diameter,  with  a  total  area  of  8.55  cm^  may  be 
employed.  The  total  thickness  of  slab  on  account  of  the  covering  for  the  steel  must 
be  increased  to  19  cm. 

From  Table  II*  for  (7^=  1000,  and  (^5  =  30,  there  is  found 


h  —  a^o.4g\^  1247  =  17.3 


cm. 


/e  =  0.00228v  12,470,000  =  8  Cm2. 

The  shear  at  the  abutment  is 

7  =  600-^1.5X1000  =  2100  kg. 

The  shearing  stress  is 

2100  ^1/9 
To  =  J  ^=1.36  kg/cm2. 

.r.J.^  .  Q-3IXI7-3' 
ioo[i7.3  — 

The  adhesion  is 


100X1.36  2 

Ti  =—-  r-^  =4-38  kg/cm2. 

9X1.1X3-14 

3.  The  floor  described  under  2  is  to  be  investigated  as  to  the  stresses  which 
occur  when  the  tensile  strength  of  the  concrete  is  taken  into  consideration. 
According  to  equation  (27),  with  the  concrete  also  acting  in  tension, 

i^i^Vi5X8.55Xi7.3 

x=  ^  =  10.02  cm., 

100X19  +  15X8.55 

*  See  Note  page  346.— (Trans-^ 


APPENDIX 


349 


and  according  to  equation  (31), 

124,700X10.02  ^ 

1 00X10.02-^    looXcS.oh'^  ^  ^.^      ^  ° 
 +  ^-^  +  15X8.55X7.282 

ig  — 10.02  ,    ,  „ 

(762  =  -X  19.4  =  17-4  kg/cm-; 

10.02 

I  c;(i7.S  — 10.02)  ,    ,  „ 

Oe=-^  '   ^X  19.4  =  21 1.4  kg/cm2. 


A  tensile  stress  of  17.4  kg/cm^  is  permissible  when  an  ultimate  tensile  strength 
of  -1X17.4  =  26.1  kg/cm^  is  demonstrated  by  experiment.  If  such  test  is  not 
feasible,  the  concrete  employed  must  show  an  ultimate  compressive  strength  of 
10X17.4  =  174  kg/cm^.  This  ultimate  compressive  strength  must  reach  180 
kg/cm^  because  of  the  assumed  permissible  stress  of  30  kg/cm^. 

To  determine  the  shearing  stress  at  the  neutral  axis,  the  distance  z  between 
the  centroids  of  tension  and  compression  must  be  found.    This  is  obtainable 

from  the  condition  that  M=Dz,  where  D^—ob=  ^^^^ ^9-4X  10.02  ^ 

2  2  ' 

so  that 

124,700 

z  =  —  =  12.83  cm. 

9720 

Then 

^^=7^o-^^^"^4kg/cm2. 

Because  of  the  tensile  strength  of  the  concrete,  the  shear  at  the  level  of  the 
reinforcement  is  somewhat  less.    In  general 

VS 

wherein  5"  is  the  statical  moment  of  the  section  above*  the  level  in  question, f 
and  /  is  the  moment  of  inertia  of  the  whole  section.  Thus,  for  the  section  of 
the  level  of  the  reinforcement, 

5  =  ioo(-^— -^^^-^  +  15X8.55X7.28  =  3698; 


so  that 


^   Mx    124,700X10.02  . 

/  =  — =  =64,420; 

Oh  19.4 

,    2100X3698  ,    ,  ^ 

To'=7  ^r^=i.2i  kg  cm2. 

64,420  X 100 


The  adhesion  is  then 


,      100X1.21        ,    ,  p 

t/  =  =4  kg/cm^. 

'     9X1.1X3.14  ^' 


*  Or  below,  as  in  the  example. — (Trans.) 

t  With  reference  to  the  neutral  axis. — (Trans.) 


350 


CONCRETE- STEEL  CONSTRUCTION 


t<  20 


4.  A  reinforced  concrete  beam  of  4  m. 
span,  with  dimensions  as  in  the  accompany- 
ing figure  has  an  apphed  moment  of  120,000 
kg. -cm.  The  maximum  compressive  stress 
in  the  concrete  and  the  stresses  in  the 
^  =  4ffO'^/m=4  5z^^''  reinforcement,  ignoring  the  tensile  stress  in 
the  concrete,  are  required. 
According  to  equation  (17) 


Fig.  8. 


14X1.51  +  15X4-52 


20 


^^^x4Xi.5i  +  i5X4.5y^A(,,xx5i  +  x5X4.5^X33)  =  ix.35cm. 


Then,  according  to  equation  (19), 

1 20,000 


(76  = 


^^^-^(33-3.78)  +  i4Xi.5xX^fx3o 
^  ■'•■^•35 


31.7  kg/cm2; 


^^-^Sf  X^''^  =  ^5o  kg/cm2; 

21. 6<  01,  o 

3-^X350  =  908  kg/cm2. 


In  computing  the  shearing  stresses,  the  distance  y\  from  equation  (24)  is  to 
be  found, 

20X11.37^ 
3 


14X8.372x1. 51 


20X11.37^ 


7.67  cm. 


+  14X8.37X1.51 


Since  the  load  per  m.  is  600  kg.,  7  =  2X600  =  1200  kg.,  and 

1200 


-^0  = 


20(21.65  +  7.67) 
20X  2.05 


=  2.05  kg/cm2; 


4X1X3-14 
At  the  upper  reinforcement,  since 


3.27  kg/cm^ 


and 


5  =  20 ^•-^  +  15X1.51X8.35  =  780, 


^  120,000X11.^^ 
/=  ^=42,970, 


31-7 

,    1200X780         ,    ,  2 

To  =  —  =  1 .00  kg/ cm^, 

°     20X42,970        ^  ^ 


APPENDIX 


351 


If  the  tensile  stresses  in  the  concrete  are  considered,  according  to  equation  (38), 
20  X  36- 

 ^  +  14(1.51X3+4.52X33) 

x  =  — —  ■  ^  =  18.8  cm., 

2oX3()  +  i4(i-5i +4.52) 

so  that,  according  to  equation  (39), 

T  20,000X18.8  ,    .  _ 

'''"'  =  20X18.83    20X17.-'  :  ~  :r'^-^  ''S/cm^; 

o  o 

'^bz=^^X2s.4  =  2i.4.  kg/cm2; 

(76  =  15X^^^X21.4  =  265  kg/cm2. 

The  shearing  stresses  at  the  level  of  the  upper  reinforcement,  since  7  =  96,410, 
will  be 

1200 /i8.82- 15.82    i5Xi.5iXi5.8\      ^   ,    ,  o 

ro=—    — +  ^      ^       ^  =0.87  kg/cm2, 

96,4io\        2  20         /        '  ^ 

and  the  adhesive  stress 

20X0.87 

Ti=  —  =  2.^  kg/cm^. 

3X0.8X3.14  ^ 

At  the  neutral  axis, 


1200  /i8.82  15X1. 51X15. 8\  ,  , 
96,4io\   2  20         /  ° 


5.  A  floor  panel  3  m.  wide  and  4  m.  long  is  to  consist  of  a  plain  concrete 
slab,  freely  supported  on  all  sides,  with  reinforcement  in  two  directions  parallel 
to  the  sides.  Live  and  dead  load  amount  to  600  kg/cm2.  The  necessary  thick- 
ness of  floor  and  amount  of  reinforcement  is  required. 

The  applied  moment,  computed  from  the  shortest  span,  is 

600X3.1^X100 
M=  =  48,050  kg. -cm. 

The  permissible  stresses  are  (Tc  =  iooo  and  ^76=40  kg/cm^.  Then,  from 
Table  II  * 


^  148,050  ^ 
h-a  =  o.s9\^-^  =8.54  cm., 

/e=o.oo293\/^4,8o5,ooo  =  6.42  cm2. 

The  thickness  of  the  slab  should  be  increased  to  10  cm.  For  the  reinforce- 
ment in  the  direction  of  the  shorter  span,  10  round  rods  of  9  mm.  diameter  with 
a  total  area  of  6.36  cm2/m.  width,  should  be  used.  The  longer  rods  can  be 
somewhat  fewer,  about  in  the  ratio  of  the  breadth  to  the  length  of  the  slab.  For 
them,  8  rods  per  m.  width,  of  the  same  size,  may  be  employed. 

*  See  note,  page  346. — (Trans.) 


352 


CONCRETE-STEEL  CONSTRUCTION 


6.  A  T-beam  of  the  dimensions  shown  in  the  accompanying  figure  is  assumed 

with  a  span  of  7.5  m.  and  a  column 
spacing  of  7.8  m.,  with  a  hve  load  of  500 
kg/m,  in  a  store.  The  reinforcement  con- 
sists of  6  round  rods  of  2.5  cm.  diameter, 
with  a  total  area  of  29.45  cm.  The  maxi- 
mum stresses  in  concrete  and  steel  are  to 
be  determined. 


 tso  

t 

/  1 

1 

[ 

a     •  • 

f 

Fig.  9. 

The  dead  load  consists  of 


The  weight  of  the  T-beam  =  (1.5  Xo.i  +0.32X0.25)  X  2400  ....  552  kg. 
The  weight  of  the  floor  filHng,  6  cm.  of  rolled  cinders  ...  36  kg. 

The  weight  of  the  cement  finish,  2  cm.  thick   40  kg. 

The  weight  of  the  plaster  ceiling   14  kg. 


Total  per  sq.  meter   90  kg. 

Thus,  for  1.5  m2,  1.5X90=   i35  kg. 

The  live  load   500  kg. 


Total   1187  kg. 

or  approximately,  1200  kg/m  length  of  beam. 
Then 

i2ooX7-82Xioo         ^  . 
M.  =  =  912,000  kg. -cm., 


and,  according  to  equation  (11), 

150X102 
2 


15X29.45X36 


150X10  +  15X29.45 
and  according  to  equation  (13) 

>'  =  i2.o5-5 


10^ 


6(2X12.05—  10) 
consequently,  according  to  equation  (14) 

912,600 


12.05  cm. 


=  8.23  cm., 


29.45(36-12.05  +  8.23) 
and  according  to  equation  (15) 

12.05 


=963  kg/cm2, 


Gh=- 


15(36-12.05) 

The  shear  at  the  abutment  is 

|r_7-5Xi2oo 


X  963  =32.3  kg/cm2. 


=  4500  kg., 


APPENDIX 


353 


so  that  the  shearing  stress  in  the  concrete  is 


V 


^0  = 


4500 


bi{h  —a—x-\-y)    25(36  —  12.05+8.23) 


=  5.6  kg/cm2. 


The  permissible  stress  is  thus  exceeded.  It  is  consequently  advisable  to  bend 
upward  near  the  end  two  rods  of  the  upper  layer  of  reinforcement.  The  point 
at  which  such  bending  should  take  place  is  determined 
by  the  condition  that  at  this  point  the  shear  V  should 
be  only 


4500X4-5 
5-6 


3616  kg. 


This  is  found  at  a  point  ^^^^ — ^^^==0.74  meters  from  the  abutment. 


The  total  tension  Z  to  be  taken  by  the  bent  portions  of  the  rods  is  equal  to 
the  shear  to  be  transferred  to  them,  i.e., 

Z  =  ^t(5.6-4.5)iX25  =  72o  kg. 

V2 

The  stress  in  the  bent  rods  is  therefore 

720 


^^=-i— —  =  73  kg/cm2. 


The  adhesive  stress  on  the  four  lower  rods  at  the  supports,  amounts  to 


25X5-6 


=  4-5  kg/cm2. 


u  4X2.5X3-14 

If  it  is  desired  to  ascertain  the  tension  in  the  concrete,  x  must  be  determined 
from  equation  (32) 

25X42^  ,  125X10^ 


+  15X29.45X36 


25X42  +  125X10  +  15X  29.45 
and  according  to  equation  (13) 


16.12  cm. 


100 


}/  =  i6.i2-5+—  -  =  11.87  cm., 

6(32.24-10) 


so  that  by  equation  (33a) 

[150X10X11. 

lf  =  9i2,6oG=  —  


^^^(2X16.12-10) +^(6.123  +  25. 


+  15X29.45X19.^ 


from  which 


,~|  Obd 
J  16^12' 


obd  =  2S.4  kg/cm2, 
2  ^  88 

<T63  =  ^1^X28.4 =45.6  kg/cm2, 

19-88      „  ,    ,  O 

<7e  =  i5X^-X28.4  =  525  kg/cm2. 


354 


CONCRETE-STEEL  CONSTRUCTION 


The  stress  abs  =  4S-^  kg/cm^  is  certainly  too  large,  so  that  the  width  of  the  stem 
of  the  beam  and  the  area  of  the  reinforcement  must  be  increased. 

7.  A  continuous  T-beam  on  four  supports  with  the  section  shown  in  the  accom- 
panying illustration  carries  500  kg/m,  in  a  store.  The  maximum  stresses  in 
the  concrete  and  the  steel  are  required.    The  weight  per  m.  length  is: 

.  (i.5Xo.io  +  o.3Xo.35)X240o=  612kg. 

To  which  is  to  be  added  the  additional  dead  load  of  the  last  example.  135  kg. 


Total. 


747  kg. 


or  approximately  750  kg/m  length  of  beam. 


600- 


500- 


5.00. 


1 — ' 

 !  

\ 

\ 

•  •  i  •  • 

•  • 

— ' 

_i 

1^  j'5  H 

Fig.  II. 


The  calculations  will  be  made  on  the  assumption  commonly  made  for  con- 
tinuous beams,  of  a  uniform  moment  of  inertia,  ignoring  the  variations  due  to' 
changes  of  size  and  location  of  reinforcement  and  of  possible  increase  of  strength 
over  the  supports.    These  points  increase  the  factor  of  safety  to  some  extent. 

The  applied  moments  are 

(a)  At  0.4  /  of  the  first  span: 

Mg=  -h  0.08X7  50X5 .02X100=  +150,000^ 
— ilfp= —0.02  X  500X5. 0"X  100=  —  25,000 
+Mp=  +0.10X500X5.0^X100=  +125,000 
Thus,  ikfniax= +275,000. 

(6)  Over  the  center  supports : 

Mg=  —0.10000X750X5.02X100=  —187,500 
—ifp=  —0.11667X500X5.02x100=  —145,838 
+lfp=  +0.01667X500X5.02x100=  +  20,838 

Thus,  ^max=  -ZZZ^ZZ^' 

(c)  In  the  center  span: 

if^= +0.025X750X5.02x100= +46,875 
— ii'p=  —0.050X500X5.02x100=  —62,500 
+ifp= +0.075X500X5.02x100=  +93,750 

Thus,  +Mn,ax=  +140,625, 

-^max=-  15^625. 

With  these  quantities  the  stresses  are  as  follows: 

{a)  At  0.4/  0}  the  First  Span. — The  reinforcement  consists  of  8  round  rods 
of  15  mm.  diameter  and  14.14  cm2  total  area  placed  5  cm.  from  the  bottom  of  the 
beam. 


APPENDIX 


355 


Since  the  neutral  axis  falls  within  the  slab,  its  location  can  be  found  by  the 
help  of  equation  (2) 


15X 14-14 
150 


/         2X150X35  1        o  ^ 

vi  H  —  —  I   =8.6^  cm. 

7       15X14.14       J  ^ 


and     are  then  given  by  equations  (4)  and  (5)  as 

2X275,000 


Ob 


150X8.63X32. 12 

275,000 
14.14X32. 12 


13.2  kg/cm2, 
606  kg/cm^. 


(h)  Over  the  Intermediate  Supports. — Since  the  concrete  can  carry  no  tensile 
stress,  only  the  rectangular  portion  of  the  section  with  the  reinforcement  diverted 
to  the  to]),  is  effective  for  the  negative  moments  over  the  supports.  Consequently, 
two  additional  rods  15  mm.  in  diameter  are  inserted  so  that  the  aggregate  area 
becomes  17.67  cm^. 

The  determination  of  the  position  of  the  neutral  axis  is  again  made  by  equation  (2) 


Oh 


Oe- 


i5Xi7-67r   I    ^  2X35X35 
35      L\'  15X17.67 

2X333,33^ 


16.66  cm. 


35X16.66X29.45 
333,33^ 


-  =  38.8  kg/cm2, 


17.67X29.45 


640  kg/ cm^. 


(c)  In  the  Central  Span. — The  maximum  positive  moment  is  considerably 
smaller  than  at  0.4/  of  the  first  span.  Four  round  rods  with  a  total  area  of 
7.07  cm^,  are  sufficient. 


X 


15X7.07 


Oh  = 


150  L 
2X140,625 


I  2X150X37.25  ^ 

\  I  i  —  I 


Oe- 


150X6.58X35-06 
140,625 


15X7.07 

=  8.1  kg/cm^, 


58  cm., 


7.07X35-06 


565  kg/cm2. 


356 


CONCRETE-STEEL  CONSTRUCTION 


One  round  rod  i  cm.  in  diameter  with  an  area  of  0.79  cm^  in  the  upper  part 
of  the  section,  is  sufficient  for  the  negative  moment  of  —15,625.  Then 


15X0.79 


Ge  = 


35 

15,625 


0-79X35-93 


^^^2X35X37.5_  1 
=  550  kg/cm2. 


If  it  is  desired  to  ascertain  in  this  case,  and  for  0.4/  of  the  first  span,  what 
is  the  tensile  stress  in  the  concrete,  there  follows. 


35X40^  115X102 
———+—-^—+15X14.14X35 

35X40  + 1 15X10  + 15X14.14 


=  14.9  cm. 


Then,  according  to  equation  (33^^), 


10.74  cm. 


Obd 

275,000=  

14.9 


^X  10X10.74(29.8-10) +^(4.93  +  25.13)4-15X14.14X20.12 

275, 000 = 2  9 ,000  Obd ; 

275,000       .    ,  ^ 

ohd^  =  9.5  kg/cm-; 

29,000     ^  ^  ^' 


2  c  I 

(76z=— —  X9.5  =  i6  kg/cm^. 
14.9 

The  determination  of  the  shearing  and  adhesive  stresses  is  made  as  in  the 
last  example. 

8.  A  reinforced  concrete  column  30X30  cm.  in  section,  with  4  round  rods 
of  16  cm^  area,  is  centrally  loaded  with  30,000  kg. 
The  induced  stresses  in  concrete  and  steel  are  to  be 
calculated. 

According  to  equations  (43)  to  (45)  there  will  result 


-50   

Fig.  14. 


30,000  =  (76(3©  X  30  + 1 5  X 1 6) ; 
30,000 


Ob 


26.3  kg/cm" 


1 140 

(7e  =  i5X26.3-395  kg/cm2. 


9.  The  same  column  is  to  be  investigated  for  buckling;  conditioned  on  its 
height  being  4  m. 
In  Euler's  formula 


APPENDIX 


357 


E  = 


2,100,000 


15 

of  safety  =  10. 


so  that 


140,000  is  the  value  assumed  for  concrete  and  5  =  the  factor 


30' 


/=^  +  i5X4X4.oXi2 


P  = 


2  =  102,060, 
I  o  X 1 40,000  X 1 02 ,060 


10  X  160,000 


•^303  kg. 


Since,  in  the  problem,  P  is  only  30,000  kg.,  no  risk  is  ex])erienced  of  the  buck- 
ling of  the  column.  In  order  that  no  buckling  should  occur  in  the  reinforce- 
ment, the  condition  must  exist  that 


Fk. 


The  stress  k  of  the  steel  has  been  found  to  approximate  395  kg/cm^.  Since, 
for  round  rods 

F  =  —    and  7= 

4  64 

it  follows  that 


and  the  permissible  length  of  rod  to  prevent  buckling  is 

4 


l  =  d' 


0X2,100,000 


80: 


395 


25. 8(/. 


1^ 


In  order  to  avoid  a  buckling  of  the  rods,  they  are  to  be  connected  by  ties  at 
distances  not  exceeding  25.8X2.26  =  58  cm.    According  to  Sec.  15,  No.  6,  the 
extreme  tie  spacing  should    not    be  greater  than 
30  cm. 

10.  A  reinforced  concrete  column  25X25  cm.  in 
section,  with  four  reinforcing  rods  of  2  cm.  diameter, 
has  a  load  of  5000  kg.  placed  eccentrically  at  a 
distance  of  10  cm.  from  the  center.  The  resulting 
concrete  and  steel  stresses  are  required. 

Two  conditions  apply  to  the  solution  of  this 
problem: 

1.  The  sum  of  the  external  and  internal  forces 
must  be  equal  to  zero ;  Sr  =  o. 

2.  The   sum   of  the   statical  moments  of  the 
forces  acting  on  a  section  must  be  zero;  Ilil/  =  o. 

Further,  the  condition  must  hold,  that  stresses*  are 
to  each  other  as  the  distances  from  the  neutral  axis 
multiplied  by  the  modulus  of  elasticity,  that  is, 

ob'.(yed=x:n{x  —  a) 
ob'.Ocz=x\n{h—a—x). 

*  Stresses  in  concrete  and  in  steel. — (Trans.) 


Fig.  15. 


358  CONCRETE-STEEL  CONSTRUCTION 

From  the  first  condition 

/  N       r>  .    r    f^~^    h-a-x\       Vbx    nfe,  ,,1 

(a)  P  =  --a,  +  nfeO,[---  __j  =  J , 

and  from  condition  2 

(b)  F{x-e)=  Ob—  +  nfeob  I  — — —  +  ^  J 


3 


Ob 


[—  +  — (2^2  —  2^:x:  +  2^2  +  7/2  —  2a//)l 
3      ^  J 


Equating  the  value  of  obtained  from  these  equations  and  reducing,  there 
results 

T^x^ — ^x^  —  (2e  —  h)x  =  2a^  +  h^  —  (2a+e)h, 
6nfe  2nfe 

or  by  substituting  the  values,  ^  =  25;  ^  =  15; /^  =  6.28;  ^  =  2.5;  h  =  2^;  a  =  3; 

x3-7.5:x;2  +  452.i6x  =  9734. 

The  solution  is  most  easily  accomplished  by  trial  and  there  is  found  with 
sufficient  accuracy 

x  =  i6.^  cm. 

Then,  by  the  aid  of  equation  (a) 

/2c;Xi6.^    1^X6.28  \ 

(76  =  20, 2  kg/cm^, 

and  further, 

Oed=—  ^  =  249  kg/cm2, 

16.3 


(7^2  =  249-^^  =  107  kg/cm2. 
13-3 


APPENDIX 


359 


Table  B 

MAXIMUM  MOMENTS  IN  CONTINUOUS  BEAMS 


CONTINUOUS  BEAMS  OF  TWO  SPANS  1:1 


X 

T 

Moments 

From  ^ 

From  p 

M 

Max.  (  +  M) 

Max.  (-M) 

+ 

o 

0 

0 

0 

O.I 

+  0.0325 

0.03875 

0.00625 

O.  2 

+  0-0550 

0.06750 

o.oi 250 

+  0.0675 

0.08625 

0.01875 

0.4 

+  0.0700 

0.09500 

0.02500 

+  0.0625 

0.09375 

0.03125 

0.6 

+  0.0450 

0.08250 

0.03750 

0.7 

+  0.0175 

0.061 25 

0.04375 

0.75 

0 

0.04688 

0.04688 

0.8 

—  0.0200 

0.03000 

0.05000 

0.85 

—  0.0425 

0.01523 

0.05773 

0.9 

— 0.0675 

0.0061 I 

0.07361 

0-95 

—  0.0950 

0.00138 

0.09638 

I 

—  0. 1250 

0 

0.1 2500 

pP 

Note. — In  this  and  the  following  table,  in  cases  in  which  the  calculations  include 
only  quiescent  loads,  as  for  roofs,  it  is  recommended  that  the  positive  moments 

at  centers  of  spans  be  increased  to  at  least  — .  In  computations  concerning  par- 
tial live  loads  p  in  unfavorable  positions,  the  deficient  value  due  to  ,.;  found  in  the 
table  is  overbalanced,  so  that  the  tabular  values  can  be  employed  as  they  stand. 

In  beams  with  more  than  four  spans,  the  end  ones  can  be  calculated  like  the 
first  one  of  a  beam  of  four  spans,  and  the  other  spans  in  the  longer  beam  like  the 
second  span  of  a  continuous  beam  of  four  spans. 


360 


CONCRETE-STEEL  CONSTRUCTION 


Table  C 

MAXIMUM  MOMENTS  IN  CONTINUOUS  BEAMS 


CONTINUOUS  BEAMS  OF  THREE  SPANS  1:1:1 


Moments 

X 

T 

From  g 

From  p 

M 

Max.  (+M) 

Max.  (-M) 

First  opening 
o 

O.I 
0.2 

0.4 

0-5 
0.6 
0.7 

0.8 

0.85 

0.9 

0.95 

I 

Second  opening 
0 

0.05 
0.1 

0-15 
0.2 

0.2764 

0-3 
0.4 

0-5 

0 

+  0-035 
+  0.060 
+  0.075 
+  0.080 
+  0.075 
+  0.060 
+  0.035 
0 

— 0.02125 
— 0.04500 
—0.07125 
— 0. 10000 

— 0. 10000 
— 0.07625 
— 0.05500 
— 0.03625 
— 0.020 
0 

+  0.005 
+  0.020 
+  0-025 

0 

0.005 

0.010 

0.015 

0.020 

0.025 

0.030 

0.035 

0.04022 

0.04898 

0.06542 

0.08831 

0.11667 

0.11667 

0.09033 

0.06248 

0.05678 

0.050 

0.050 

0.050 

0.050 

0.050 

+ 
0 

0.040 

0.070 

0.090 

0. 100 

0. 100 

0.000 

0.070 

0.04022 

0.02773 

0.02042 

o.oi 706 

0.01667 

0.01667 

0.01408 

0.00748 

0.02053 

0.030 

0.050 

0-055 
0.070 
0.075 

pP 

pP 

APPENDIX 


361 


Table  D 

MAXIMUM  MOMENTS  IN  CONTINUOUS  BEAMS 


CONTINUOUS  BEAMS  OF  FOUR  SPANS  1:1:1:1 


Moments 

X 

T 

From  g 

From  p 

M 

Max.  {-M) 

Max.  (+M) 

First  opening 
o 

O.l 
0.2 

0-3 
0.4 
0.5 
0.6 
0.7 

0.7857 
0.8 
0.85 
0.9 

0-95 
I  0 

Second  opening 
0 

0.05 
0.1 

0-15 
0.2 
0. 2661 

0-3 
0.4 

0.5 
0.6 
0.7 
0.8 

0.8053 

0.85 

0.9 

0.95 
1 .0 

0 

+  0.03429 
+  0.05857 
+  0.07286 
+  0.07714 
+  0.07143 
+  0.05572 
+  0.03000 
0 

-0.00571 

—  0.02732 

—  0.05143 

—  0.07803 
— 0. 10714 

— 0 . 10714 
—0.08160 
-0.05857 

—  0.03803 

—  0.02000 

0 

+  0.00857 
+  0.02714 
+  0.03572 
+  0.03429 
+  0.02286 
+  0.00143 
0 

—0.01303 
— 0.03000 
—0.04947 

—  0.07143 

0 

0.00536 
0.01071 
0.01607 
0.02143 
0.02679 
0.03214 
0.03750 
0.04209 
0.04309 
0.05216 
0.06772 
0.09197 
0.12054 

0. 1 2054 
0.09323 
0.0721 2 
0.06340 
0.05000 
0.04882 
0.04821 
0.04643 
0.04464 
0.04286 
0.04107 
0.04027 
0.04092 
0.04754 
0.06105 
0.08120 
0. 10714 

+ 
0 

0.03964 
0.06929 
0.08893 
0-09857 
0.09822 
0.08786 
0.06750 
0.04209 
0.03738 
0.02484 
0.01629 
0.01393 
0.01340 

0.01340 
0. 01 163 
0.01455 
0.02537 
0.03000 
0.04882 
0.05678 
0.07357 
0.08036 
0.07715 
0.06393 
0.04170 
0.04092 
0.03451 
0.03105 
0.03173 
0.03571 

pP 

pP 

362 


CONCRETE-STEEL  CONSTRUCTION 


Table  E 

CONVERSION  TABLE,  METRIC  TO  ENGLISH 


No. 

Kilograms 
to  Averdupois 
Pounds. 

Tonnes  to 
Tons  of 
2000  Pounds. 

Centimeters 
to  Inches. 

Meters  to 
Feet. 

oQU3,re 
Centimeters 
to  Square 
Inches. 

Square 
Meters 
to  Square 
Feet. 

I 

2. 20462 

I . 10231 

0-39370 

3-280833 

0-155 

10.76387 

2 

4.40924 

2.20462 

0.78740 

6.561667 

0.310 

21-52773 

3 

6.61387 

3-30693 

1 .18110 

9.842500 

0.465 

32. 29160 

4 

8.81849 

4.40924 

1.57480 

13-123333 

0.620 

43-05547 

5 

II .02311 

5-51156 

I .96850 

16.404167 

0-775 

53-81934 

6 

13-22773 

6.61387 

2.36220 

19.685000 

0.930 

64.58320 

7 

15-43236 

7.71618 

2-75590 

22.965833 

1.085 

75-34707 

8 

17.63698 

8.81849 

3.14960 

26. 246667 

1 . 240 

86.11C94 

9 

19.84160 

9.92080 

3-54330 

29-527500 

1-395 

96.87481 

No. 

Cubic  Meters  to 
Cubic  Yards. 

Hektoliters 
to  Bushels. 

Kilograms  per 

Square 
Centimeters  to 
Pounds  per 
Square  Inch. 

Kilograms  per 
Square  Meter 
to  Pounds  per 
Square  Foot. 

Kilograms  per 
Cubic  Meter  to 
Pounds  per 
Cubic  Foot. 

I 

1-30794 

2.83774 

14.22340 

0. 20482 

0.06243 • 

2 

2.61589 

5-67548 

28.44680 

0.40963 

0. 12486 

3 

3-92383 

8-51323 

42.67020 

0.61445 

0.18728 

4 

5-23177 

11-35097 

56.89359 

0.81927 

0. 24971 

5 

6-53971 

14.18871 

71 .11699 

I .02408 

0.31214 

6 

7.84766 

17.02645 

85-34039 

I . 22890 

0.37457 

7 

9-15560 

19.86420 

99-56379 

1-43372 

0.43700 

8 

10.46354 

22. 70194 

113. 78719 

1.63854 

0.49943 

9 

II. 77149 

25-53968 

128.01059 

1-84335 

0.56185 

INDEX 


PAGE 

^DHESION   42 

proof  of  existence  of   2 

Adhesive  forces,  formulas  for   145,  146 

stresses  developed   157,  161,  184,  186 

Anchorage  for  ends  of  rods   148,  149 

Arch  bridges,  examples   266-284 

Arches   14 

in  buildings  ,   235 

gEAMS,  graphical  methods  of  calculating   130-137 

of  variable  depth,  shear  in   190,  191 

Bending  and  tensile  strengths,  explanation  of  discrep- 
ancy between   26,  27,  28 

relation  of   27-29 

of  rods,  effects  of   181,  182 

strength  of  concrete   25 

tests  to  determine  extensibility   55?  56 

with  axial  compression,  formula  for   1 19-123 

with  axial  stress,  graphical  methods  for   132-137 

with  axial  tension   127-129 

with  axial  thrust   1 19-127 

Beton  Frette   67 

Brackets   8,  9 

Breaking  strength  of  columns   63 

Brick  curtain  walls   215 

Buildings  of  reinforced  concrete,  examples  of   210 

/CANTILEVERS,  examples   257 

Cellars,  Water-tight   247 

Cement  bins,  examples   295-298 

Coal  pockets,  examples   290 

Coefficient  of  expansion,  concrete   3 

steel   3 


363 


364  INDEX 

PAGE 

Column  tests   60,  6 1 

Columns   11 

breaking  strength  of   63 

effect  of  ties  on   62 

flexure  of   65 

with  longitudinal  rods   59 

with  spiral  reinforcement    67 

Compression   59 

Compressive  strength  of  concrete   19,  20 

Computation,  suggested  methods  of   323,  333,  340 

Concrete,  bending  strength  of   25 

compressive  strength  of   19 

elasticity  of   21 

extensibility  of   50 

punching  resistance  of   31 

shearing  strength  of   31 

tensile  strength  of   20 

torsional  strength  of   39?  40 

Continuous  bridges,  examples   263-266 

T-beams,  experiments  with   199-203 

slabs   5,  6 

Cooling  towers,  example  of   3'^3~3^5 

Corrosion,  proof  of  prevention  of   1,2 

Cracking  of  T-beams   188-190 

stage,  methods  of  computation  for   137,  138 

Cracks,  safety  against  tension   105,  106,  107 

Curtain  walls  of  brick   215 

J^EFLECTION  of  beams   192,  193 

of  Isar-Grimwald  arch  liridge   280 

Deformation   192,  193 

Deformed  bars,  objections  against   i7ji8 

Distributing  rods   5 

Domes   235 

Double  reinforcement,  formulas  for   87,  88 

for  slabs,  with  methods  of  figuring   94 

tests  of  slabs  with   93 

JgFFECTIVE  depth,  alteration  of,  from  cracks   171 

Effect  of  humidity  on  extensibility  of  reinforced 

concrete   53~5^ 

Elasticity  of  concrete   21 

Elastic  limit  of  concrete   21 

Examples  of  methods  of  computation   326-333 

Expanded  metal,  objections  against   17 

Expansion  by  heat,  coefficient  of,  concrete   3 

coefficient  of,  steel   3 


INDEX 


365 


PAGE 

Extensibility   50 

from  bending  tests   55 j  56,  57 

of  reinforced  concrete    51 5  52 

Euler's  formula  not  applicable  to  long  columns   65 


J^IREPROOF     quality    of     reinforced  concrete, 

example  of   212 

Flexure  of  columns   65 

Floor  systems   209 

Footings   241 


r^OVERNMENT  (Prussian)  regulations   233 

Grain  bins,  examples   304-306 

Graphical  methods  of  calculating  beams   130-137 

Grimwald-Isar  arch  bridge   271-280 


TJAUNCHES   6,  8,  9 

in  continuous  beams,  effect  of   203 

History   204,  205 

Hooks,  effects  of   158,  161,  162,  164,  178,  182 

Humidity,  effect  of  on  concrete   4 


JINTERIOR    treatment    of    reinforced  concrete 

buildings   224 

Isar-Grimwald  arch  bridge   271-280 


J^NEES   8,  9,  10 

J^EITSATZE   15,317,333 

Longitudinal  and  spiral  reinforcement  for  columns  72,73 

rods,  effects  of   64 


jyjELAN  arch  bridges   281-283 

Methods  of  computation,  suggested   323,  333,  340 

Modulus  of  elasticity  of  concrete   21-25 

Moments,  computation  of   194-197 

in  slabs   6 


^^EUTRAL  axis,  position  of   95-98 

^^RE  pockets,  examples   290,  293,  298 

piLE  formula   255 

Piles   250 

Prussian  government  regulations  for  reinforced  con- 
crete work   16 


366 


INDEX 


t 


PAGE 

Pumice  concrete   19 

Punching  resistance  of  concrete   31 


■REGULATIONS  of  Prussian  government   16 

Reinforcing  rods,  arrangement  in  columns  ...  . .  12 

arrangement  of,  in  arches   11 

arrangement  of,  in  slabs   5,  6,  9,  10 

Remant  stresses   198 

Reservoirs,  examples   284-287 

Rich  mixture,  necessity  of   18 

Roof  construction   230 

coverings   219-230 

Rust,  proof  of  prevention  of   2 


OAFE  stresses,  suggested   323-339 

Safety  against  tension  cracks   105-107 

Sand-boxes  for  Isar  Grimwald  arch  bridge   276 

Scherfestigkeit   31?  32 

Schubfestigkeit   31^32 

Shearing  forces,  effects  of   139,  141,  147,  148 

experiments  concerning,  by  author. . . .  151-173 
experiments   concerning  at  Stuttgart 

Testing  Laboratory   173,  174 

formulas  for   141-146 

theory  of  action  of  ,   151 

strength  of  concrete   3^^  33.  34,  35>  3^,  37»  3^ 

from  slotted  beams   40,41,42 

relation  of  tensile  and  compressive  to..    32,  33,  35,  42 

stresses   31,32 

in  beams  of  variable  depth   190,  191 

in  reinforced  concrete   31,  32,  55 

Shipping  platform,  example  of   312 

Shrinkage  stresses   198 

Simple  bending   74 

Silos,  computation  of  stresses  in   306-312 

examples   287-312 

Slab  culverts,  examples   256 

Slabs   4 

formulas  for  computation  of   77-82 

graphical  computation  of   76 

tables  for  computation  of   S3-86 

tests  of  simple   90 

tests  of,  with  double  reinforcement   93 

with  double  reinforcement,  methods  of  figuring  94 

Sliding  resistance  of  rods  in  concrete   43,  44,  45,  46,  47 

Slope  of  bent  rods   7 


INDEX 


367 


Spiral  and  longitudinal  reinforcement  for  columns 
reinforcement  for  columns  

notes  regarding  

for  columns,  how  computed  

Spirally  reinforced  columns  


PAGE 

73 

231 

71 

67 

47 

74 

7=^ 

232 

16 
8 

149^ 

150 

Stirrups  

action  of  

effects  of   155,  156,  158,  159,  164, 

168, 175,  177,  178,  180, 
181,  183,  184,  188 

on  adhesion   48,  49 

tests  concerning  effect  of   152 

Stress  distribution  over  section   98-104 

Stresses  at  failure   172-179 

in  T-beams   11 2-1 1 3 

for  slabs,  working   92 

Sunken  well  casings   244 

Suspension  system  of  reinforcement,  tests  concerning. .  164,  165,  166,  168 

Systems  of  construction   206-209 


npABLES  for  beams  subjected  also  to  thrust  

T-beam  bridges,  examples  

T-beams  

continuous,  experiments  with  

cracking  of  

diagrams  for  

economical  spacing  of  

experiments  concerning,  by  Stuttgart  Labora- 
tory   

formulas  for  

stresses  in   

Tensile  strength  of  concrete  

Tension  bending  with  axial  

Tests  of  spirally  reinforced  columns  

Thacher  rods,  spHtting  effect  of  

Theory,  development  of  

Thrust,  axial  bending  with  

Ties  in  columns,  effect  of  ,. , 

Torsional  strength  of  concrete 

Torsion  tests  on  hollow  cylinder  

Truss  action  of  reinforcement .  .;.  /-J-l' 

theory  of  .  J.  1 

Tunnel  lining,  example  of  


125,  126, 127 
258-266 
8-107 
1Q9-203 
188-190 
115 


176 

108-110,  114,  116,  117 
112 
20 

127,  128,  129 
68,  69 

47 
208 

119-127 

53-.  54,  55  ,^ 

160.  ;  16 1  ; , :  /• 

161,  ''i83,'i8-;,'i«8^ 

312  ,  .  . 


36S 


INDEX 


PAGE 

■^^^ARIABLE  depth,  shear  in  beams  of   190,  191 

Tl/'ATERTTGHT  cellars   247 

Weld,  location  of   11 

Well  casings   244 

Wet  mixture,  strength  of   18-20 

Working  stresses  for  slabs   92 


f 


GETTY  CENTER  LIBRARY 


3  3125  00002  1424 


